When students are asked to reduce fractions like 144/216 or find prime factorizations, setting up long division for every potential divisor wastes valuable time. Divisibility rules are mathematical inspection shortcuts that tell you with 100% certainty whether a number divides evenly by another—without performing any division at all. Below is the ultimate classroom reference guide for divisors 2 through 10.
The Master Divisibility Cheat Sheet
1. Divisibility by 2 (Even Numbers)
The Rule: The last digit must be even: 0, 2, 4, 6, or 8.
Example: 3,498 ends in 8, so it is divisible by 2. 7,125 ends in 5, so it is not.
2. Divisibility by 3 (Sum of Digits)
The Rule: Add all the digits together. If the sum is a multiple of 3, the entire number is divisible by 3.
Example: Test 5,421:
5 + 4 + 2 + 1 = 12. Since 12 is divisible by 3 (12 ÷ 3 = 4), 5,421 is divisible by 3!
3. Divisibility by 4 (Last Two Digits)
The Rule: Look only at the last two digits. If the number formed by the last two digits is divisible by 4, the entire number is divisible by 4.
Why: Because 100 is divisible by 4 (100 ÷ 4 = 25), all hundreds, thousands, and higher places are always divisible by 4.
Example: Test 9,536. Look at 36: 36 ÷ 4 = 9. Therefore, 9,536 is divisible by 4.
4. Divisibility by 5 (0 or 5)
The Rule: The last digit must be 0 or 5.
Example: 8,245 and 930 are divisible by 5; 412 is not.
5. Divisibility by 6 (Both 2 and 3)
The Rule: The number must satisfy BOTH the rule for 2 and the rule for 3 (it must be even AND its digits must sum to a multiple of 3).
Example: Test 732:
- Is it even? Yes (ends in 2).
- Do digits sum to a multiple of 3? 7 + 3 + 2 = 12 (Yes).
- Therefore, 732 is divisible by 6!
6. Divisibility by 8 (Last Three Digits)
The Rule: Check if the last three digits form a number divisible by 8.
Why: Because 1,000 is divisible by 8 (1,000 ÷ 8 = 125).
Example: 5,128 → check 128: 128 ÷ 8 = 16. Divisible!
7. Divisibility by 9 (Sum Equals 9)
The Rule: Sum all digits. If the sum is divisible by 9, the original number is divisible by 9.
Example: Test 48,357:
4 + 8 + 3 + 5 + 7 = 27. Since 27 is divisible by 9 (27 ÷ 9 = 3), 48,357 is divisible by 9!
8. Divisibility by 10 (Ends in 0)
The Rule: The units digit must be 0.
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Frequently Asked Questions
Because 10 divided by 9 leaves a remainder of 1 (10 = 9+1, 100 = 99+1), each power of 10 contributes its coefficient to the remainder. For 4, 100 is divisible by 4, so only the last two digits matter. For 6, a number must satisfy both rule 2 and rule 3 simultaneously.
Examine only the last three digits. If the hundreds digit is even, check if the last two digits are divisible by 8. If the hundreds digit is odd, add 4 to the last two digits and check if that sum is divisible by 8.
Yes! Divisibility tests are the primary tool for reducing fractions to their lowest terms. Testing numerator and denominator for 2, 3, and 5 quickly uncovers common factors without brute-force division.