Divisibility Tests

Number Theory · Axiom Academy

Learn quick methods to test divisibility and understand why they work using modular arithmetic Test for 2: A number is divisible by 2 if and only if its last digit is even (0, 2, 4, 6, or 8). Test for 5: A number is divisible by 5 if and only if its last digit is 0 or 5. Any number can be written as where d is the last digit. Since 10 is divisible by both 2 and 5, we have and . Therefore, the divisibility of the entire number depends only on the last digit d . Test for 3: A number is divisible by 3 if and only if the sum of its digits is divisible by 3. Example: 1,458 → 1+4+5+8 = 18 → 18 is divisible by 3 ✓ Proof Using Modular Arithmetic: Consider a number with digits: Therefore, the number is divisible by 3 if and only if the sum of digits is divisible by 3. Test for 9: A number is divisible by 9 if and only if the sum of its digits is divisible by 9. Example: 7,362 → 7+3+6+2 = 18 → 18 is divisible by 9 ✓ Similar to the test for 3, we use . Since all powers of 10 are congruent to 1 (mod 9), the number has the same remainder as the digit sum when divided by 9. Test for 4: A number is divisible by 4 if and only if its last two digits form a number divisible by 4. Example: 3,516 → Last two digits: 16 → 16 ÷ 4 = 4 ✓ Any number can be written as where the last two digits form the number L . The divisibility by 4 depends only on the last two digits. Test for 11: A number is divisible by 11 if and only if the alternating sum of its digits is divisible by 11.

This is the written version of the interactive lesson above. See the full Number Theory course.