Divisibility Rules • Topic 1 of 2

Rules for 2 to 11

These are the rules every CAT aspirant should fire on sight. By 2: the last digit is even. By 5: the last digit is 0 or 5. By 10: it ends in 0. By 4: the number formed by the LAST TWO digits is divisible by 4 (because 100 is). By 8: the LAST THREE digits are divisible by 8 (because 1000 is). By 3 and by 9: add all the digits — if that sum is divisible by 3 or by 9, so is the number; you can keep adding until a single digit remains (digital root). By 6: it must pass both the 2-test and the 3-test. By 11: take the alternating sum of digits (right to left, +, −, +, −, …); if the result is 0 or a multiple of 11, the number is divisible by 11. The 7-rule is fiddly, so for 7 most CAT students just divide or use known patterns. Speed tip: never test 3 by the digit sum and forget evenness when the divisor is 6 or 12.

✅ Solved examples

1. Is 4,38,156 divisible by 4?
Look only at the last two digits: 56. Since 56 = 4 × 14, the whole number is divisible by 4. Yes.
2. Is 91,728 divisible by 8?
Check the last three digits: 728. 728 ÷ 8 = 91 exactly, so the number is divisible by 8. Yes.
3. Is 7,29,063 divisible by 9?
Digit sum = 7+2+9+0+6+3 = 27, and 27 is divisible by 9. So yes, the number is divisible by 9 (and hence by 3).
4. Is 8,16,948 divisible by 11?
Alternating sum from the right: 8 −4 +9 −6 +1 −8 = 0. A result of 0 means it is divisible by 11. Yes.

✏️ Practice — try these, take hints as needed

1. Is 5,43,672 divisible by 8?
Use only the last three digits.
Test 672.
672 ÷ 8 = 84.
Yes (672 = 8 × 84)
2. Is 2,34,561 divisible by 3 but not by 9?
Add the digits.
Sum = 21.
21 ÷ 3 = 7, but 21 is not a multiple of 9.
Yes — divisible by 3, not by 9
3. Is 7,42,816 divisible by 4?
Last two digits only.
Test 16.
16 ÷ 4 = 4.
Yes
4. Is 9,02,351 divisible by 11?
Alternating sum from the right.
1 −5 +3 −2 +0 −9.
= −12, not 0 or ±11.
No (alternating sum = −12)
5. Is 1,11,111 divisible by 6?
It must pass both the 2-test and the 3-test.
Last digit is 1 (odd).
Fails the 2-test.
No (it is odd)

📝 Topic test — 8 questions

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