Divisibility, remainders and units digits — Maths Olympiad practice
Number theory questions look hard but have neat shortcuts: divisibility tests, remainders from the nearest multiple, and the fact that the units digits of powers repeat in short cycles.
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How to solve it
Patterns repeat — “divisibility tests and cycles of last digits save long calculations”
Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11.
Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder.
Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.
Use it for: Divisibility, remainder, last-digit and “smallest number to add” questions in Class 6–10.
Worked example: What is the units digit of 7^45? → 7
The units digits of powers of 7 repeat: 7, 9, 3, 1 — a cycle of 4.
45 ÷ 4 = 11 remainder 1, so 7^45 ends like 7^1, which ends in 7.
Check: 7^4 = 2401 ends in 1, so 7^44 ends in 1 and 7^45 = 7^44 × 7 ends in 7 ✓
✏️ Practice
Work it out in your head or on paper, type the answer and press Check. Stuck? Press “Show me how”.
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🏁 Topic 19 test
20 questions. Check each answer, then go to the next one. Get 16 or more right to pass.
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