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Divisibility rule

A divisibility rule is a shorthand method in number theory for determining whether a given integer is divisible by a specific divisor without carrying out the complete division algorithm.^[1] These rules leverage the properties of the decimal (base-10) number system and modular arithmetic to simplify checks, particularly for small prime divisors like 2, 3, 5, 7, and 11, as well as their composites such as 4, 6, 8, 9, and 10.^[2] By reducing a large number to a smaller equivalent form—such as the sum of its digits or its last few digits—the rules determine divisibility if and only if the reduced form is divisible by the target number, avoiding lengthy computations.^[3] The foundation of these rules lies in congruence relations from modular arithmetic, where a number n in base 10 can be expressed as n = a_k \cdot 10^k + \cdots + a_1 \cdot 10 + a_0, and powers of 10 modulo the divisor reveal patterns.^[4] For instance, since $10 \equiv 1 \pmod{3}, the entire number n is congruent to the sum of its digits modulo 3, making the sum-of-digits rule effective for 3 and 9 (where $10 \equiv 1 \pmod{9}).^[2] Similarly, $10 \equiv 0 \pmod{2} and \pmod{5} explains why divisibility by 2 or 5 depends solely on the last digit, while for 11, the alternating sum arises from $10 \equiv -1 \pmod{11}.^[2] More complex rules, like those for 7 or 13, involve iterative processes but follow the same modular principles.^[5] Common divisibility rules for small integers are summarized below, illustrating their practical application in mental arithmetic and computational efficiency: These rules not only aid in factorization and prime testing but also extend to applications in cryptography, error detection in codes (e.g., ISBN checks using modulo 11), and simplifying fractions by identifying common factors.^[7] While rules exist for larger numbers, they grow more intricate, often requiring computational tools beyond manual methods.^[2]

Fundamentals

Definition and Purpose

A divisibility rule is a heuristic method for determining whether a given positive integer is evenly divisible by another integer, meaning the division results in no remainder, without requiring the complete execution of the division algorithm.^[8] These rules typically rely on the decimal representation of the number, leveraging properties such as the parity of its last digit, the sum of its digits, or patterns in its trailing digits to assess divisibility.^[9] For example, to determine if 123 is divisible by 3 without performing the division, a rule can be applied to quickly confirm the condition based on the number's digits.^[10] The primary purpose of divisibility rules is to enhance computational efficiency, particularly in mental mathematics and when handling large numbers where full division would be time-consuming or impractical.^[11] They also serve as valuable tools for error-checking in arithmetic calculations, allowing verification of results by confirming expected divisibility properties.^[12] Furthermore, these rules offer foundational insights into number theory, providing a "window" into the intrinsic properties and behaviors of integers.^[13] Divisibility rules require a prerequisite understanding of integers—whole numbers with no fractional or decimal parts—and the basic principle that one integer divides another if the remainder of the division is zero.^[9] Specific rules for small divisors, such as those from 2 to 10, are often introduced first in educational contexts due to their simplicity and utility.^[10]

Historical Context

The origins of divisibility rules trace back to ancient mathematical traditions, with the earliest documented specific test appearing in the Babylonian Talmud around 500 CE, which included a method for checking divisibility by 7 to determine positions in the Sabbatical Cycle.^[14] In ancient Greece, Euclid's Elements (c. 300 BCE) established foundational principles of number theory in Books VII–IX, discussing divisibility, greatest common divisors, and concepts akin to modular arithmetic without explicit digit-based tests.^[15] During the medieval period, Islamic scholars advanced these ideas, with Persian mathematician Abu Bakr al-Karaji (d. after 1019) employing checks for divisibility by 9 and 11 to verify computational accuracy in his algebraic works.^[16] These methods, rooted in modular properties, influenced broader arithmetic practices. In 1202, Italian mathematician Leonardo of Pisa (Fibonacci) introduced divisibility rules for 7 and 11 to Europe in his Liber Abaci, drawing from Arabic sources and adapting them for the Hindu-Arabic numeral system to facilitate commerce and calculation.^[17] The 19th century saw a formalization of divisibility within systematic number theory, as exemplified in Carl Friedrich Gauss's Disquisitiones Arithmeticae (1801), which rigorously explored congruences and divisibility properties, laying groundwork for modern analytic approaches. By the 20th century, divisibility rules became standard in school curricula worldwide to enhance arithmetic efficiency, appearing in elementary texts as practical tools for mental computation. British biologist and mathematician Lancelot Hogben further popularized these rules in his 1943 edition of Mathematics for the Million, presenting them accessibly to lay audiences to demystify everyday numeracy.

Rules for Small Divisors

Divisors 1–10

The simplest divisibility rules apply to the divisors from 1 to 10 and are particularly easy to use because they rely on examining one or a few digits of the number or simple combinations thereof, exploiting the structure of the base-10 system where powers of 10 have predictable relationships with these divisors.^[18] For the divisor 1, every integer is divisible by 1, as 1 is the multiplicative identity and divides any number without remainder. This holds trivially in any base, including base 10.^[18] For the divisor 2, a number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). This rule works because 10 is divisible by 2 in base 10, making the divisibility of the entire number dependent solely on the last digit.^[19] For the divisor 3, a number is divisible by 3 if the sum of its digits is divisible by 3. In base 10, this is possible because 10 ≡ 1 (mod 3), so each power of 10 contributes the digit itself to the total modulo 3, reducing the number to the sum of its digits modulo 3.^[20] For the divisor 4, a number is divisible by 4 if the number formed by its last two digits is divisible by 4. The rationale stems from base 10 where 100 ≡ 0 (mod 4), so the last two digits determine the remainder, as higher powers of 10 are multiples of 100 and thus of 4.^[21] For the divisor 5, a number is divisible by 5 if its last digit is 0 or 5. This follows from 10 being divisible by 5 in base 10, meaning the last digit alone controls the divisibility.^[19] For the divisor 6, a number is divisible by 6 if it is divisible by both 2 and 3. Since 6 = 2 × 3 and 2 and 3 are coprime, the base-10 rules for each combine directly to check both conditions.^[18] For the divisor 7, a practical rule involves taking the last digit of the number, doubling it, and subtracting the result from the number formed by the remaining digits; the original number is divisible by 7 if the result is divisible by 7 (repeat the process if the result is large). This method arises from the base-10 property that 10 × 2 ≡ -1 (mod 7) after adjustment, allowing iterative reduction while preserving divisibility.^[1]^[3] For the divisor 8, a number is divisible by 8 if the number formed by its last three digits is divisible by 8. In base 10, 1000 ≡ 0 (mod 8), so the last three digits capture the remainder, with higher digits contributing multiples of 1000.^[21] For the divisor 9, a number is divisible by 9 if the sum of its digits is divisible by 9. Similar to the rule for 3, this holds because 10 ≡ 1 (mod 9) in base 10, making the number congruent to the sum of its digits modulo 9.^[22] For the divisor 10, a number is divisible by 10 if its last digit is 0. This is due to 10 dividing 10 exactly in base 10, requiring the units place to contribute no remainder.^[19] These rules form the foundation for checking divisibility in everyday calculations and can be applied iteratively for efficiency.

Divisors 11–30

Divisibility rules for integers from 11 to 30 exhibit greater complexity than those for smaller divisors, frequently requiring multi-step digit manipulations or leveraging combinations of rules for prime factors, particularly for composite numbers. These rules preserve divisibility properties while reducing the number to a smaller form that can be checked more easily. For composite divisors in this range, such as 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, and 30, the standard approach is to verify divisibility by each of the divisor's prime factors simultaneously, building on simpler rules for 2, 3, 5, 7, and 11.^[1]^[23] The rule for 11 states that a number is divisible by 11 if the alternating sum of its digits (starting with addition for the rightmost digit) is divisible by 11, including 0.^[24] For 12, the number must be divisible by both 3 (sum of digits divisible by 3) and 4 (last two digits form a number divisible by 4).^[24] For 13, a multi-step process involves taking the last digit, multiplying it by 4, and adding the result to the number formed by the remaining digits; the original number is divisible by 13 if this new number is.^[23] The rule for 14 requires divisibility by both 2 (last digit even) and 7 (a separate multi-step check often needed). For 15, the number must be divisible by both 3 and 5 (last digit 0 or 5).^[1] Divisibility by 16 is determined by checking if the number formed by the last four digits is divisible by 16.^[11] For 17, remove the last digit and subtract five times that digit from the remaining number; the result must be divisible by 17.^[23] The rule for 18 combines divisibility by 2 and 9 (sum of digits divisible by 9).^[1] For 19, multiply the last digit by 2 and add it to the remaining digits to form a new number; if this is divisible by 19, so is the original.^[23] Divisibility by 20 requires the last digit to be 0 and the last two digits to form a number divisible by 4. For 21, check divisibility by both 3 and 7.^[1] The rule for 22 is divisibility by both 2 and 11. For 23, multiply the last digit by 7 and add it to the remaining digits; the result must be divisible by 23.^[23] For 24, the number must be divisible by both 3 and 8 (last three digits divisible by 8).^[1] A number is divisible by 25 if its last two digits are 00, 25, 50, or 75.^[11] For 26, check divisibility by both 2 and 13. The rule for 27 involves a multi-step process, such as repeatedly applying a transformation like multiplying the last digit by 19 and adding to the rest until a small number is obtained.^[1] For 28, verify divisibility by both 4 and 7. For 29, multiply the last digit by 3 and add it to the remaining digits; the original is divisible by 29 if this result is.^[23] Finally, divisibility by 30 requires the number to be divisible by 2, 3, and 5 simultaneously.^[1] These composite rules underscore the efficiency of decomposing into prime factors, reducing the need for unique procedures in many cases.^[23]

Application Examples

Even and Power-of-10 Divisors

Divisibility by 2 is determined by examining the last digit of the number; if it is even—specifically 0, 2, 4, 6, or 8—the entire number is divisible by 2.^[25] For instance, consider 2468: its last digit is 8, which is even, confirming divisibility by 2 (2468 ÷ 2 = 1234). This rule functions intuitively because in base 10, any number can be expressed as 10k + d, where d is the last digit; since 10 is divisible by 2, the parity of the number matches the parity of d, meaning even endings yield even numbers.^[26] For divisibility by 4, which is 2², the rule requires checking if the number formed by the last two digits is divisible by 4. To apply this, isolate the last two digits and perform the division: for 1236, the last two digits are 36, and 36 ÷ 4 = 9 (an integer), so 1236 is divisible by 4 (1236 ÷ 4 = 309). This step-by-step process—extracting the tens and units digits, forming the two-digit number, and verifying divisibility—avoids full division of the original number.^[27] A number is divisible by 5 if its last digit is 0 or 5. For example, 1250 ends in 0, making it divisible by 5 (1250 ÷ 5 = 250). In contrast, numbers ending in digits 1 through 9 except 5, such as 1234 (ending in 4), are not divisible by 5, as their remainders when divided by 5 are nonzero.^[28] The rule for 8, or 2³, involves the last three digits: if the number they form is divisible by 8, so is the whole number. Applying this to 1024, the last three digits are 024 (equivalent to 24), and 24 ÷ 8 = 3 (an integer), confirming 1024 is divisible by 8 (1024 ÷ 8 = 128). The process entails isolating the hundreds, tens, and units digits, forming the three-digit number (padding with leading zeros if needed), and checking its divisibility by 8 through direct division or known multiples.^[29] Divisibility by 10 requires the number to end in 0, indicating it is a multiple of 10. For 100, which ends in 0, it is divisible by 10 (100 ÷ 10 = 10); this ties directly to trailing zeros, as each zero represents a factor of 10.^[30] These rules extend to higher powers of 2. For 16 (2⁴), check if the last four digits form a number divisible by 16; for 4096, the last four digits are 4096 itself, and 4096 ÷ 16 = 256 (an integer), so it is divisible. Similar patterns apply to 32 (2⁵, last five digits) and beyond, generalizing the check to the appropriate number of trailing digits based on the power.^[31] A common pitfall arises when confusing rules for powers of 2: for example, 14 is divisible by 2 (last digit 4 is even) but not by 4 (last two digits 14 ÷ 4 = 3.5, not an integer).^[32] Such combinations also underpin rules for numbers like 6 (divisible by both 2 and 3), 12 (by 4 and 3), and 20 (by 4 and 5).^[10]

Sum and Alternating Sum Divisors

Sum and alternating sum divisibility rules provide efficient ways to test divisibility for specific divisors by manipulating the digits of a number rather than performing long division. These methods stem from the fact that powers of 10 have particular remainders when divided by the divisor, allowing digit operations to preserve congruence modulo that divisor. Building on the basic sum rule for 3 from smaller divisors, these techniques extend to related numbers like 9 and 11, and can be adapted for higher values such as 27 through grouped sums.^[18] For divisibility by 3, compute the sum of all digits in the number; if this sum is divisible by 3, the original number is as well. Consider the number 123: the digits sum to $1 + 2 + 3 = 6, and since $6 \div 3 = 2 with no remainder, 123 is divisible by 3. For much larger numbers, such as 999999, the initial sum is $9 + 9 + 9 + 9 + 9 + 9 = 54; although 54 is divisible by 3 ($54 \div 3 = 18), further reducing it via $5 + 4 = 9 (also divisible by 3) confirms the result efficiently. The rule for 9 is analogous but stricter: the sum of the digits must be divisible by 9. For 81, $8 + 1 = 9, and $9 \div 9 = 1, so 81 is divisible by 9. In contrast, for 82, $8 + 2 = 10, and reducing further to $1 + 0 = 1 (not divisible by 9) shows that 82 is not. This highlights how the rule distinguishes multiples precisely through the digital root process.^[33] For 11, form the alternating sum of the digits, typically starting from the rightmost digit as positive. If this alternating sum is divisible by 11 (including 0, 11, -11, 22, -22, etc.), the number is divisible by 11. For 121, the alternating sum is $1 - 2 + 1 = 0, which is divisible by 11, confirming 121 ($11 \times 11) is. Negative results or zeros are handled by taking the absolute value or recognizing 0 as valid; for larger alternating sums, apply the rule recursively until manageable.^[34] A sum-based approach for 27, a higher power of 3, involves dividing the digits into groups of three from the right (padding with leading zeros if needed) and summing these three-digit blocks, since $10^3 = 1000 \equiv 1 \pmod{27}. The original number is divisible by 27 if this block sum is. For the small number 27, treat it as 027; the sum is 27, and $27 \div 27 = 1, so yes. For illustration with a two-digit case like 27, an ad hoc computation of the sum of cubes of digits yields $2^3 + 7^3 = 8 + 343 = 351, and $351 \div 27 = 13 exactly, though this is not a general rule and applies only coincidentally to certain small multiples. More reliably, for larger examples like 2644272, group as 002, 644, 272; sum $2 + 644 + 272 = 918, and $918 \div 27 = 34, confirming divisibility.^[18] The multi-iteration process is essential for very large numbers in these rules: repeatedly apply the digit sum (for 3 and 9) or block sum (for 27) or alternating sum (for 11) until reaching a single digit or small number that can be checked directly against the divisor. This reduces computational effort while preserving the modulo property; for instance, the digital root for 3 or 9 must be 0, 3, 6, or 9 (for 3) or 9 (for 9). These rules also serve as tools for error checking in arithmetic and calculator operations, particularly via the "casting out nines" method tied to divisibility by 9. In multiplication, for example, the digital root of the product should equal the digital root of the product of the factors' digital roots modulo 9; mismatches indicate errors like transposition or miscalculation. This detects common mistakes in about 90% of single-digit errors without full recomputation.

Other Specific Divisors

The divisibility rule for 7 involves taking the last digit of the number, doubling it, and subtracting the result from the remaining digits; if the outcome is divisible by 7 (including 0), the original number is divisible by 7, and the process can be repeated for larger results. For example, consider 343: the last digit is 3, doubled to 6; subtract from the rest (34 - 6 = 28); since 28 ÷ 7 = 4 with no remainder, 343 is divisible by 7. For a larger number like 1234567, apply the rule iteratively: last digit 7 doubled to 14, 123456 - 14 = 123442; next, 2 doubled to 4, 12344 - 4 = 12340; then 0 doubled to 0, 1234 - 0 = 1234; 4 doubled to 8, 123 - 8 = 115; 5 doubled to 10, 11 - 10 = 1; since 1 is not divisible by 7, 1234567 is not divisible by 7. For 13, one method is to add four times the last digit to the remaining number; if the result is divisible by 13, so is the original.^[23] Applying this to 169: last digit 9, four times 9 is 36; add to the rest (16 + 36 = 52); 52 ÷ 13 = 4 with no remainder, so 169 is divisible by 13.^[23] An alternative approach subtracts nine times the last digit from the rest; for 1234: last digit 4, nine times 4 is 36; 123 - 36 = 87; 87 ÷ 13 ≈ 6.692 (remainder 9), so 1234 is not divisible by 13. The rule for 17 requires subtracting five times the last digit from the remaining number; divisibility holds if the result is divisible by 17.^[23] For 289: last digit 9, five times 9 is 45; 28 - 45 = -17; -17 ÷ 17 = -1 with no remainder, confirming 289 is divisible by 17.^[23] For a larger example like 12383: last digit 3, five times 3 is 15; 1238 - 15 = 1223; repeat with 3 (five times 15), 122 - 15 = 107; 107 ÷ 17 ≈ 6.294 (remainder 5), so 12383 is not divisible by 17.^[23] For 19, add twice the last digit to the remaining number; if divisible by 19, the original is as well, though variations like adding four times the last two digits exist for some cases.^[23] Using 361: last digit 1, twice 1 is 2; 36 + 2 = 38; 38 ÷ 19 = 2 with no remainder, so 361 is divisible by 19.^[23] For 1147: last digit 7, twice 7 is 14; 114 + 14 = 128; 128 ÷ 19 ≈ 6.737 (remainder 14), indicating 1147 is not divisible by 19.^[23] When handling large numbers with these rules, breaking the number into groups of three digits from the right and applying alternating addition and subtraction can enhance efficiency, particularly for 7, 11, and 13, as 1000 ≡ -1 modulo these divisors because 1001 = 7 × 11 × 13 is a multiple of each.^[23] For instance, with the seven-digit number 123456789 and the rule for 7, group as 123,456,789: compute 123 - 456 + 789 = 456; since 456 ÷ 7 ≈ 65.143 (not integer), it is not divisible, avoiding step-by-step processing of all digits.^[23] This grouping reduces computational steps compared to iterative single-digit methods.^[23] These multiplier-based rules offer significant efficiency over direct long division, especially for mental arithmetic or large numbers, by reducing the problem to smaller equivalents in fewer operations and enabling quick checks without full quotients or remainders.^[12] For composite divisors like 21, the rule combines checks for 3 (sum of digits) and 7 (multiplier method).

Rules for Larger Divisors

Composite Divisors Beyond 30

For composite divisors greater than 30, the standard approach involves factorizing the divisor into its prime or smaller composite factors and then applying the established divisibility rules for each factor sequentially; a number is divisible by the composite if it satisfies all the individual rules.^[35]^[36] This method leverages the multiplicative property of divisibility, where if n is divisible by p and q (with p and q coprime or not), then n is divisible by pq.^[11] Consider 33, which factors as $3 \times 11. To test divisibility by 33, first check if the sum of the digits is divisible by 3, and then verify if the alternating sum of the digits is divisible by 11. For example, 363: the digit sum is $3 + 6 + 3 = 12 (divisible by 3), and the alternating sum is $3 - 6 + 3 = 0 (divisible by 11), so 363 is divisible by 33.^[37] For 35, factored as $5 \times 7, the number must end in 0 or 5 (for divisibility by 5) and then be tested using the rule for 7 (multiply the last digit by 2 and subtract from the rest, checking if the result is divisible by 7). An example is 245: it ends in 5, and $24 - 2 \times 5 = 14 (divisible by 7), confirming divisibility by 35.^[38]^[11] The rule for 39, which is $3 \times 13, requires the digit sum to be divisible by 3 and the number to satisfy the divisibility test for 13 (add 4 times the last digit to the remaining number and check divisibility by 13). For 507: digit sum $5 + 0 + 7 = 12 (divisible by 3), and $50 + 4 \times 7 = 78 (78 ÷ 13 = 6), so 507 is divisible by 39.^[11]^[39] Divisibility by 45 ($5 \times 9) is determined by the number ending in 0 or 5 and the digit sum being divisible by 9. Testing 405: ends in 5, digit sum $4 + 0 + 5 = 9 (divisible by 9), thus divisible by 45.^[32] For 50 ($2 \times 25), the last two digits must be 00 or 50, extending the rules for 2 (even) and 25 (last two digits 00, 25, 50, or 75). For instance, 150 ends in 50, so it is divisible by 50.^[40]^[41] The test for 75 ($3 \times 25) combines the last two digits being 00, 25, 50, or 75 with the digit sum divisible by 3. Example: 225 ends in 25, digit sum $2 + 2 + 5 = 9 (divisible by 3), confirming divisibility by 75.^[42]^[40] Combining rules for composite divisors is generally more efficient than deriving or applying a standalone rule for the full composite, as it reuses simple, well-known tests for smaller factors, reducing computational steps for manual checks.^[36]^[35] However, for very large composites with multiple or high prime factors, the combined tests can become cumbersome if individual prime rules are complex, in which case grouping factors into intermediate composites (e.g., checking for 15 before 105) or resorting to modular arithmetic may be suggested for practicality.^[18]^[11]

Prime Divisors Beyond 30

For prime divisors greater than 30, divisibility rules typically involve more complex manipulations of digits, such as multipliers applied to the last digit (or last few digits) or grouping digits into blocks of two or three, reflecting the increasing distance of these primes from powers of 10. These methods leverage congruences modulo the prime, often requiring iterative application for large numbers to reduce the size until a direct check is feasible. While effective for mental arithmetic or quick verification, such rules grow in complexity as the prime increases, prioritizing efficiency over simplicity compared to rules for smaller divisors. The divisibility rule for 31 uses a multiplier method: remove the last digit, subtract three times that digit from the remaining number, and check if the result is divisible by 31 (repeating if necessary for larger results). For example, consider 1247: remove 7, subtract $3 \times 7 = 21 from 124 to get $124 - 21 = 103; since $103 \div 31 \approx 3.32 (not integer), 1247 is not divisible by 31.^[43] For 37, a grouping method applies due to $10^3 \equiv 1 \pmod{37}: divide the number into groups of three digits starting from the right, sum these groups, and test if the sum is divisible by 37. For instance, with 999, the single group is 999, and $999 \div 37 = 27 (exact), confirming divisibility; for larger numbers like 74,037, groups are 074 and 037, summing to 111, and $111 \div 37 = 3 (exact). This approach exploits 37's factorization of repunits like 111 ($111 = 3 \times 37).^[44] The rule for 41 involves a similar multiplier: remove the last digit, subtract four times that digit from the remaining number, and check divisibility by 41 iteratively. For a three-digit example like 123: remove 3, subtract $4 \times 3 = 12 from 12 to get $12 - 12 = 0 (divisible by 41), so 123 is divisible by 41 ($123 \div 41 = 3). This can be extended recursively for longer numbers.^[45] For 43, add thirteen times the last digit to the remaining number and test the result for divisibility by 43. An example is 2157: remove 7, add $13 \times 7 = 91 to 215 to get $215 + 91 = 306; since $306 \div 43 \approx 7.12 (not integer), 2157 is not divisible by 43. This multiplier derives from solving for a suitable congruence factor.^[46] The rule for 47 requires subtracting fourteen times the last digit from the remaining number, checking if the result is divisible by 47. For 236: remove 6, subtract $14 \times 6 = 84 from 23 to get $23 - 84 = -61; since -61 \div 47 \approx -1.3 (not integer), 236 is not divisible by 47. For larger numbers, iteration applies, often involving adjustments for negative results.^[47] As primes exceed 30, patterns emerge in these rules, with increasing dependence on two- or three-digit groupings (e.g., for 37 via $10^3) or multipliers approximating the modular inverse of 10 modulo the prime, such as those found by solving $10k \equiv 1 \pmod{p}. For very large primes, these manual rules become cumbersome and lengthy, often outperformed by computational methods like direct modular reduction using algorithms such as long division or binary exponentiation for $10^k \mod p. Such rules for primes can underpin composite tests by verifying prime factors, but standalone application remains focused on the prime itself.^[48]

General Methods

Generalized Divisibility Rule

The generalized divisibility rule offers a systematic framework for constructing divisibility tests for any positive integer divisor d based on the base-10 representation of a number and properties of modular arithmetic. Consider a positive integer n expressed as n = a \times 10^k + b, where b consists of the last k digits of n and a is the number formed by the remaining leading digits. Then, n \equiv 0 \pmod{d} if and only if a \times 10^k + b \equiv 0 \pmod{d}. By precomputing $10^k \mod d, the condition can be handled as follows: if d divides $10^k (so $10^k \equiv 0 \pmod{d}), then n \equiv b \pmod{d}, and divisibility requires b \equiv 0 \pmod{d}. If \gcd(10^k, d) = 1, the condition can be recast as checking whether a linear combination a + m b \equiv 0 \pmod{d}, where m is the modular inverse of $10^k \mod d or another convenient multiplier that simplifies the expression while preserving congruence.^[9] This approach leverages the specific properties of base 10, where powers of 10 often yield simple residues modulo small d, enabling concise rules. For example, since $10 \equiv 1 \pmod{9}, all powers $10^i \equiv 1 \pmod{9}, so n \equiv sum of its digits \pmod{9}; thus, n is divisible by 9 (or by 3, as $3 \mid 9) if the digit sum is. Likewise, $10 \equiv -1 \pmod{11}, so even powers of 10 are \equiv 1 \pmod{11} and odd powers \equiv -1 \pmod{11}, leading to the rule that n is divisible by 11 if the alternating sum of its digits is.^[49] To derive a rule for a given d, compute the sequence of powers $10^i \mod d for i = 0, 1, 2, \dots until a pattern or cycle emerges, as the powers are periodic with period dividing \phi(d) by Euler's theorem if \gcd(10, d) = 1. For n = \sum_{i=0}^{m} d_i 10^i with digits d_i, this yields n \equiv \sum_{i=0}^{m} d_i (10^i \mod d) \pmod{d}, so divisibility holds if the weighted digit sum is \equiv 0 \pmod{d}. Grouping digits or using repeating cycles simplifies application for larger n. As an illustration for d=7, note that $10 \equiv 3 \pmod{7}. For a two-digit number n = 10a + b, n \equiv 3a + b \pmod{7}. An equivalent form is a - 2b \equiv 0 \pmod{7}, derived by adjusting coefficients since $3(a - 2b) = 3a - 6b \equiv 3a + b \pmod{7} (as -6 \equiv 1 \pmod{7}) and 3 is invertible modulo 7. For longer numbers, apply the process recursively to the result.^[3] The method generalizes beyond base 10 to any base b > 1, by replacing powers of 10 with powers of b modulo d to obtain weighted sums for the digits in that base. For instance, in base 2, powers of 2 modulo an odd d allow rules based on bit positions, often simplifying to checks on the number of 1s or parity in certain subsets.^[50] For large d, the rules become cumbersome due to lengthy cycles in $10^i \mod d or irregular weights, rendering them impractical for manual checks. They remain efficient, however, in computational settings where modular exponentiation and reductions can be performed algorithmically.^[9]

Alternative Techniques

Alternative techniques for divisibility testing encompass methods that deviate from conventional digit-based rules, often relying on decomposition, estimation, modular computations, or computational optimizations suitable for specific contexts such as large numbers or irregular divisors. These approaches can be particularly useful when standard rules are inefficient or unavailable, such as for prime divisors beyond simple patterns or very large integers where full computation is feasible but optimized checks are preferred.^[51] One such method involves factorization, where the dividend is broken down into its prime factors, and each factor of the divisor is checked for presence in the dividend's factorization with sufficient multiplicity. For instance, to determine if 12 divides 360, factor 12 as $2^2 \times 3 and verify that 360 = $2^3 \times 3^2 \times 5 contains at least two 2s and one 3; since it does, 12 divides 360 evenly. This technique is effective for composite divisors with known factorizations, especially in number theory applications, though it becomes computationally intensive for large primes without efficient factoring algorithms. A shortcut using partial quotients in long division allows estimation of the remainder without completing the full process. In this approach, the dividend is divided by subtracting multiples of the divisor iteratively, recording partial quotients until the remainder is less than the divisor; divisibility holds if the final remainder is zero. For example, dividing 1234 by 17 begins with 17 × 7 = 119 (partial quotient 7, subtract from 123 leaving 4, bring down 4 to make 44), then 17 × 2 = 34 (partial quotient 2, remainder 10), and continuing yields a total quotient of 72 with remainder 10, indicating non-divisibility. This method builds number sense and is an alternative to traditional long division for mental or quick checks on moderately sized numbers.^[52] Casting out nines, primarily a check for divisibility by 9 or 3 via digit sums, extends to other moduli like 7 through modular tables based on powers of 10 modulo the divisor. For 7, since $10 \equiv 3 \pmod{7}, $10^2 \equiv 2 \pmod{7}, $10^3 \equiv -1 \pmod{7}, and $10^4 \equiv -3 \pmod{7}, one multiplies each digit by the corresponding residue (repeating every four digits: 1, 3, 2, -1) and sums the results; the original number is divisible by 7 if this weighted sum is. Applying to 1234: $4 \times 1 + 3 \times 3 + 2 \times 2 + 1 \times (-1) = 4 + 9 + 4 - 1 = 16, and 16 mod 7 = 2 (not 0), confirming non-divisibility. This tabular extension aids verification for divisors without simple digit rules.^[53] In computer science, binary or ternary checks leverage bit-shift operations for divisibility by powers of 2, as right-shifting bits by k positions effectively divides by $2^k and checks if the result is an integer (no fractional bits lost). For example, to test if 16 ( $2^4 ) divides 48 in binary (110000), shifting right by 4 bits yields 0011 (3), an integer, confirming divisibility; for 50 (110010), shifting gives 0011.001 (non-integer), indicating it does not. This is efficient in hardware or programming for powers of 2, avoiding arithmetic overhead.^[54]^[55] The polynomial analogy views a decimal number as a polynomial evaluation n = f(10) = a_k 10^k + \cdots + a_0, where divisibility by d occurs if f(10) \equiv 0 \pmod{d}; to test, substitute a root r where $10 \equiv r \pmod{d} and check if f(r) \equiv 0 \pmod{d}, providing a framework for custom rules. This conceptual tool underpins many tests but serves as an alternative lens for irregular divisors by computing f(r) directly rather than full division.^[56] These techniques are especially applicable for irregular divisors lacking mnemonic rules or very large numbers where digit summation is cumbersome, such as testing primality-related divisibility in cryptography. Historically, pre-calculator tools like the abacus facilitated divisibility by performing successive divisions through bead manipulations to observe exact quotients, while slide rules enabled quick division approximations to verify integer results without full computation.^[51]^[57]^[58]

Mathematical Foundations

Algebraic Proofs

Algebraic proofs of divisibility rules rely on expressing a positive integer n in its decimal form as n = \sum_{k=0}^{m} d_k 10^k, where each digit d_k satisfies $0 \leq d_k \leq 9, and then manipulating this expansion to factor out the divisor or show equivalence to a simpler expression divisible by the same number. This approach uses properties of powers of 10 and polynomial identities to demonstrate when n is divisible by a given integer d without performing full division. Such derivations highlight how the structure of base-10 representation simplifies checks for small divisors. For divisibility by 3 and 9, observe that $10^k - 1 = 999\ldots9 (with k nines), which equals $9 \times 111\ldots1 and is thus divisible by 9 for any k \geq 1. Rearranging gives $10^k = 1 + 9 \times (10^{k-1} + 10^{k-2} + \cdots + 1), where the parenthetical term is an integer. Substituting into the expansion of n, we obtain n = \sum_{k=0}^{m} d_k + 9 \times \sum_{k=1}^{m} d_k (10^{k-1} + \cdots + 1). The second sum is an integer multiple of 9, so n minus the sum of its digits is divisible by 9. Therefore, n is divisible by 9 if and only if the sum of its digits is divisible by 9. Since 3 divides 9, the same logic shows that n is divisible by 3 if and only if the sum of its digits is divisible by 3.^[59]^[60] For divisibility by 11, consider the difference $10^k - (-1)^k, which is divisible by 11 for all k \geq 0. This holds for k=0 since $1 - 1 = 0, and for k=1 since $10 - (-1) = 11. By induction, assume it holds for some k; then for k+1, $10^{k+1} - (-1)^{k+1} = 10 \times 10^k + (-1)^k = 10 \times [10^k - (-1)^k] + 11 \times (-1)^k, both terms of which are multiples of 11. Thus, $10^k = (-1)^k + 11 \times q_k for some integer q_k. Substituting into n yields n = \sum_{k=0}^{m} d_k (-1)^k + 11 \times \sum_{k=0}^{m} d_k q_k. The second sum is an integer multiple of 11, so n minus the alternating sum of its digits (with signs (-1)^k) is divisible by 11. Hence, n is divisible by 11 if and only if this alternating sum is divisible by 11.^[59] Divisibility by 2, 5, or 10 depends solely on the last digit because higher powers of 10 incorporate these factors. For 2, write n = 10a + b where b is the last digit; since $10a = 2 \times 5a is even, n is even if and only if b is even. For 5, $10a = 5 \times 2a is divisible by 5, so n is divisible by 5 if and only if b = 0 or $5. For 10, combining the above, n is divisible by 10 if and only if b = 0.^[1]^[60] For 4, 8, and 25, the last two or three digits suffice due to divisibility of higher powers of 10. Since $10^2 = 100 = 4 \times 25, any n = 100a + r (where r is the last two digits) has $100a divisible by 4, so n is divisible by 4 if and only if r is. For 8, $10^3 = 1000 = 8 \times 125, so writing n = 1000a + s ( s the last three digits) shows n divisible by 8 if and only if s is. For 25, again $100 = 4 \times 25, so n = 100a + r is divisible by 25 if and only if r = 00, 25, 50, or $75.^[60]^[1] For 7 and 13, algebraic rearrangements allow checking via modified forms of the number. Consider n = 10a + b for 7; multiplying by 5 gives $5n = 50a + 5b = 7 \times 7a + (a + 5b), so $5n - (a + 5b) is divisible by 7. Since 5 and 7 are coprime (gcd(5,7)=1), n is divisible by 7 if and only if a + 5b is. Similarly, for 13, multiplying by 4 yields $4n = 40a + 4b = 13 \times 3a + (a + 4b), so $4n - (a + 4b) is divisible by 13; since gcd(4,13)=1, n is divisible by 13 if and only if a + 4b is. These multipliers derive from solving $10m - 1 divisible by the divisor (e.g., $10 \times 5 - 1 = 49 = 7 \times 7; $10 \times 4 - 1 = 39 = 3 \times 13). Alternatively, rearranging as n - 10a = b and iterating shows rules like subtracting twice the last digit from the rest for 7.^[1]

Modular Arithmetic Proofs

Modular arithmetic provides a rigorous framework for proving divisibility rules in base 10, leveraging congruences to simplify checks for specific divisors. For powers of 2 and 5, the rules arise from the factorization of 10 as $2 \times 5, which ensures that sufficiently high powers of 10 are congruent to 0 modulo these divisors. Consider divisibility by $2^n. A number N in base 10 can be expressed as N = a \cdot 10^k + b, where b is the last k digits and a is the remaining prefix, with k \geq n. Since $10^k = (2 \cdot 5)^k = 2^k \cdot 5^k, it follows that $10^k \equiv 0 \pmod{2^n} for k \geq n, because $2^k is divisible by $2^n. Thus, N \equiv b \pmod{2^n}, so N is divisible by $2^n if and only if b is divisible by $2^n. For example, modulo 4 (n=2), $10^2 = 100 \equiv 0 \pmod{4}, confirming the last two digits determine divisibility.^[2] The proof for powers of 5 is analogous. For $5^n, $10^k = 2^k \cdot 5^k \equiv 0 \pmod{5^n} when k \geq n, as $5^k divides $5^n. Therefore, N \equiv b \pmod{5^n}, and divisibility of N by $5^n depends solely on the last n digits b. For instance, divisibility by 25 (n=2) requires checking the last two digits, since $10^2 = 100 \equiv 0 \pmod{25}. This symmetry between 2 and 5 stems from their shared factors with the base 10.^[2] For prime divisors like 7, where \gcd(10,7)=1, a recursive rule can be derived using the modular inverse of 10. Consider N = 10a + b, where b is the last digit and a is the rest. Then N \equiv 0 \pmod{7} if and only if $10a + b \equiv 0 \pmod{7}. Since $10 \equiv 3 \pmod{7} and the inverse of 3 modulo 7 is 5 (as $3 \cdot 5 = 15 \equiv 1 \pmod{7}), but for the standard rule of doubling the last digit and subtracting, note that N = 10(a - 2b) + 21b. Here, $21 \equiv 0 \pmod{7}, so N \equiv 10(a - 2b) \pmod{7}. Because \gcd(10,7)=1, $10x \equiv 0 \pmod{7} implies x \equiv 0 \pmod{7}. Thus, N \equiv 0 \pmod{7} if and only if a - 2b \equiv 0 \pmod{7}, allowing recursive application to a - 2b.^[2] An extension to 13 follows a similar approach. For N = 10a + b, $10 \equiv 10 \pmod{13}, and the inverse of 10 modulo 13 is 4, since $10 \cdot 4 = 40 \equiv 1 \pmod{13} (as $40 - 3 \cdot 13 = 1). Multiplying the congruence $10a + b \equiv 0 \pmod{13} by 4 yields $40a + 4b \equiv 0 \pmod{13}, or a + 4b \equiv 0 \pmod{13}. Therefore, N is divisible by 13 if and only if a + 4b is, enabling a rule where the last digit is multiplied by 4 and added to the remaining number, with recursion for larger values. The powers of 10 modulo 13 cycle (e.g., $10^2 = 100 \equiv 9 \equiv -4 \pmod{13}), but the inverse-based method generalizes grouping rules.^[18] In general, for any prime p \neq 2,5, a divisibility rule exists because \gcd(10,p)=1, so 10 has a modular inverse modulo p. This allows constructing rules via the inverse multiplier or by finding the order of 10 modulo p, often using continued fractions for efficient digit groupings or the minimal polynomial of 10 over \mathbb{Z}/p\mathbb{Z}. Base-10 arithmetic facilitates these rules for odd primes excluding 5 due to coprimality with 10, enabling reduction to smaller equivalents without loss of divisibility information.^[3]

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