Number TheoryFoundation6 min read#divisibility#number theory#mental math
Divisibility Rules That Save Exam Time
Knowing why 3, 9, 11 and 7 work turns a memorised trick into a tool you can rebuild in ten seconds — including for bases and moduli nobody taught you.
Olympiad Desk · Core Learning
What you will take away
- Digit rules come from 10 ≡ 1 (mod 3) and 10 ≡ −1 (mod 11)
- Alternating digit sums test divisibility by 11
- For 7 and 13, chunk the number and use 1001 = 7 · 11 · 13
- Build a rule for any modulus by tracking 10^k mod m
Every digit rule is a consequence of how powers of ten behave modulo the divisor — which means you can derive a rule for any modulus you meet, instead of hoping one was taught.
- mod 3 and mod 9: since 10 ≡ 1, the number ≡ its digit sum
- mod 11: since 10 ≡ −1, the number ≡ its alternating digit sum
- mod 7 and mod 13: since 1001 = 7 · 11 · 13, chunk digits into blocks of three and add with alternating signs
Worked problems
- 1
Determine the digit d for which 5d34 is divisible by 9.
Show hint
Divisibility by 9 is a statement about the digit sum.
Show solution
The digit sum is 5 + d + 3 + 4 = 12 + d, which must be a multiple of 9. The only digit making 12 + d divisible by 9 is d = 6, giving 18.
Answer: d = 6.
