Divisibility Rules Every Candidate Should Know
Learn divisibility rules for 2, 3, 4, 5, 8, 9 and 11, why each works, and how to check composite divisors, with practice questions.
Expected Interview Answer
Divisibility rules let you determine whether a number divides another exactly by inspecting a small, derived property of its digits — such as the last digit, digit sum, or an alternating digit sum — instead of performing full division.
Each rule targets a specific structural property: divisibility by 2, 4, 5, 8 depends only on the last one, two, or three digits because those powers of 10 interact predictably with the divisor; divisibility by 3 and 9 depends on the digit sum because 10 ≡ 1 (mod 3) and 10 ≡ 1 (mod 9); divisibility by 11 depends on the alternating digit sum because 10 ≡ -1 (mod 11). For composite divisors like 6 or 12, apply the rules for their coprime factors (2 and 3, or 4 and 3) simultaneously. These rules turn a division problem into fast mental arithmetic, which is exactly why they matter under interview time pressure.
- Answers divisibility questions in seconds without dividing
- The same modular logic explains every rule, so none need rote memorization
- Composite divisors reduce to combining rules for coprime factors
AI Mentor Explanation
Umpires check whether a team’s total overs bowled hits an exact boundary (like a full 50-over quota) just by checking the balls-per-over count without recounting every single delivery from ball one. Divisibility rules work the same way — checking whether a number is divisible by 4 only requires looking at its last two digits, a quick structural check standing in for full division, exactly the umpire’s shortcut of checking a small marker instead of the whole innings.
Worked example
Divisible by 8?
- last 3 digits: 368
- 368 ÷ 8 = 46 → yes
Divisible by 9?
- digit sum = 27
- 27 ÷ 9 = 3 → yes
Divisible by 72?
- 8 and 9 coprime
- both pass → yes
Step-by-Step Explanation
Step 1
Identify the divisor type
Prime power of 2/5 (last digits) vs digit-sum type (3, 9) vs alternating type (11).
Step 2
Apply the matching rule
Last n digits for 2^n/5^n; digit sum for 3/9; alternating digit sum for 11.
Step 3
Split composite divisors
For 6, 12, 15, etc., check the coprime prime-power factors separately.
Step 4
Confirm all factors pass
The number is divisible by the composite only if every coprime factor check passes.
What Interviewer Expects
- Correct rule recall for 2, 3, 4, 5, 8, 9, 11
- Understanding why each rule works (the modular reasoning), not just memorization
- Correct decomposition of composite divisors into coprime factors
- Fast, division-free mental checks under time pressure
Common Mistakes
- Applying the digit-sum rule to divisors like 4 or 8 where it does not apply
- Forgetting to alternate signs correctly for the divisibility-by-11 rule
- Splitting a composite divisor into non-coprime factors (e.g., 4 and 6 for 12 instead of 4 and 3)
- Checking only one factor of a composite divisor and assuming that is sufficient
Best Answer (HR Friendly)
“Each divisibility rule comes from how powers of 10 behave under a given modulus. Last-digit rules work for 2 and 5 because those divide 10 directly; digit-sum rules work for 3 and 9 because 10 leaves remainder 1; the alternating-sum rule for 11 comes from 10 leaving remainder -1. For a composite number like 12, I just check its coprime factors, 4 and 3, separately.”
Follow-up Questions
- Why does the last-digit rule work for divisibility by 2 and 5 but not 3?
- How would you derive a divisibility rule for 7?
- How do you check divisibility by 12 correctly using coprime factors?
- Why must the two factors used for a composite divisor check be coprime?
MCQ Practice
1. Which number is divisible by both 3 and 4 (hence by 12)?
1,236: digit sum = 12 (div by 3); last 2 digits 36 (div by 4). Both pass, so divisible by 12.
2. Is 918,082 divisible by 11? (alternating sum from the right)
From right: 2-8+0-8+1-9 = -22, a multiple of 11, so yes it is divisible by 11.
3. A number is divisible by 9 exactly when?
Since 10 ≡ 1 (mod 9), a number and its digit sum always share the same remainder mod 9.
Flash Cards
Divisibility rule for 4? — Last two digits form a number divisible by 4.
Divisibility rule for 9? — Digit sum is divisible by 9.
Divisibility rule for 11? — Alternating digit sum (right to left) is a multiple of 11.
How to check divisibility by 12? — Check divisibility by 4 and by 3 separately (coprime factors).