Divisibility, Factors & Multiples • Topic 1 of 5

Divisibility Rules

Divisibility tests let you decide whether one number divides another without doing the division. The high-value ones: a number is divisible by 2 if its last digit is even; by 3 if the sum of its digits is a multiple of 3; by 4 if its last two digits form a multiple of 4; by 5 if it ends in 0 or 5; by 9 if its digit sum is a multiple of 9; and by 6 if it passes both the 2-test and the 3-test at once. For composite divisors, split into coprime parts and test each — divisibility by 12 means divisible by 3 and by 4. These tests turn a slow calculator entry into a two-second mental check, which matters because the GRE calculator is deliberately basic. The digit-sum tests (3 and 9) are the ones test-takers most often forget, yet they appear constantly.

✅ Solved examples

1. Is 5,742 divisible by 3?
Digit sum = 5 + 7 + 4 + 2 = 18, which is a multiple of 3. So yes, 5,742 is divisible by 3.
2. Is 1,236 divisible by 4?
Check only the last two digits: 36. Since 36 = 4 × 9, the whole number is divisible by 4. Yes.
3. The number 3,4_6 is divisible by 9. What is the missing digit?
Digit sum = 3 + 4 + d + 6 = 13 + d must be a multiple of 9. The nearest is 18, so d = 5. The number is 3,456.
4. Quantitative Comparison. n is divisible by 6. Quantity A: the remainder when n is divided by 3. Quantity B: the remainder when n is divided by 2. Which is greater?
Divisible by 6 means divisible by both 2 and 3, so both remainders are 0. Quantity A = Quantity B = 0; they are equal.

✏️ Practice — try these, take hints as needed

1. Is 8,514 divisible by 9?
Add the digits.
8 + 5 + 1 + 4 = 18.
18 is a multiple of 9.
Yes
2. Is 2,730 divisible by 6?
Test 2 and 3.
Ends in 0 → even.
Digit sum 12 → multiple of 3.
Yes
3. What digit d makes 47d divisible by 5?
Divisible by 5 rule.
Ends in 0 or 5.
d = 0 or 5.
0 or 5
4. Is 1,000,001 divisible by 3?
Digit sum.
1 + 0 + ... + 1 = 2.
2 is not a multiple of 3.
No
5. Is 936 divisible by 12?
Test 3 and 4.
Digit sum 18 → by 3.
Last two digits 36 → by 4.
Yes

📝 Topic test — 8 questions

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