What Are Divisibility Tests?
A number n is divisible by another number k when n ÷ k produces no remainder. For example, 24 is divisible by 6 because 24 ÷ 6 = 4 with zero remainder, but 24 is not divisible by 5 because the result is 4.8.
Rather than performing division every time, divisibility rules are mathematical tricks that reveal divisibility instantly by examining specific digits or patterns. These rules exploit properties of our base-10 number system and the factors of various integers.
Rules exist for all common divisors:
- Simple rules based on the last digit (for 2, 5, 10)
- Rules checking the sum of all digits (for 3, 9)
- Rules examining the last two or three digits (for 4, 8, 25)
- More sophisticated alternating-sum techniques (for 7, 11, 13)
Quick Rules for Powers of 2 and 5
The simplest divisibility tests examine only the final digits of a number, since powers of 2 and 5 depend entirely on how a number ends.
Divisibility by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). Example: 138 ends in 8, so it's divisible by 2.
Divisibility by 4: Check whether the last two digits form a number divisible by 4. Example: 316 has last two digits 16; since 16 ÷ 4 = 4, the number 316 is divisible by 4.
Divisibility by 8: Check whether the last three digits form a number divisible by 8. Example: 5,432 has last three digits 432; since 432 ÷ 8 = 54, the number 5,432 is divisible by 8.
Divisibility by 5: A number is divisible by 5 if its last digit is 0 or 5. Example: 2,965 ends in 5, so it's divisible by 5.
Divisibility by 10: A number is divisible by 10 if it ends in 0. More generally, divisibility by 10n requires the last n digits to be zero.
Digit Sum Rules for 3 and 9
These rules rely on a remarkable property of base-10 representation: any number shares the same remainder with its digit sum when divided by 3 or 9.
Sum of digits = d₁ + d₂ + d₃ + ... + dₙ
If (Sum of digits) mod 3 = 0 → Number is divisible by 3
If (Sum of digits) mod 9 = 0 → Number is divisible by 9
d₁, d₂, d₃, ..., dₙ— Individual digits of the number from left to rightmod— Modulo operation (remainder after division)
Advanced Rules for 7, 11, and 13
Divisibility by 7: Subtract twice the last digit from the remaining number. If the result is divisible by 7, so is the original. Example: 343 → 34 − 2(3) = 28, and 28 ÷ 7 = 4, so 343 is divisible by 7.
Divisibility by 11: Compute the alternating sum of digits, starting from the right: subtract the last digit, add the next, subtract the next, and so forth. If this alternating sum is divisible by 11, so is the original number. Example: 1,243 → 3 − 4 + 2 − 1 = 0, and 0 is divisible by 11, so 1,243 is divisible by 11.
Divisibility by 13: Group the number into blocks of three digits from right to left, then compute the alternating sum of those blocks. Example: 9,111,414 → 414 − 111 + 9 = 312. Since 312 ÷ 13 = 24, the original number is divisible by 13.
These rules work because 7, 11, and 13 have special relationships with powers of 10 (e.g., 10³ ≡ −1 mod 13), making alternating sums and digit manipulations reveal divisibility patterns.
Common Pitfalls and Best Practices
Avoid these mistakes when applying divisibility rules:
- Negative results from alternating sums — When computing alternating sums for divisibility by 11 or 13, you may get a negative number. This is fine—simply ignore the negative sign and check if the absolute value is divisible by your divisor. For example, an alternating sum of −33 means the number is divisible by 11 because 33 is.
- Confusing digit count with digit value — For divisibility by 4 and 8, you examine the last two or three digits as a complete number, not individual digits. For 728, the last two digits form 28, and you check if 28 is divisible by 4—not whether 2 and 8 individually are.
- Forgetting to repeat for large numbers — If an alternating sum or reduced number is still very large, apply the rule again. For instance, checking divisibility of 987,654 by 11 may yield an alternating sum like 143—you'd then apply the rule again to confirm if 143 itself is divisible by 11.
- Mixing up rules for similar divisors — Rules for 3 and 9 both use digit sums, but divisibility by 3 requires the sum to be divisible by 3, whereas divisibility by 9 requires divisibility by 9. Similarly, rules for 2, 4, and 8 all examine the last digits but in different quantities (1, 2, and 3 respectively).