Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 20 · 40 questions

How to solve it (Patterns repeat): 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.

1. 1.What is the smallest number you must add to 116 to make it divisible by 3? (a) 1 (b) 2
2. 2.What is the remainder when 177 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
3. 3.What is the remainder when 178 is divided by 3? (a) 0 (b) 2 (c) 1
4. 4.Which of these numbers is divisible by 6? (a) 2004 (b) 2012 (c) 2009 (d) 2002
5. 5.Which of these numbers is divisible by 3? (a) 2432 (b) 2434 (c) 2428 (d) 2430
6. 6.What is the units digit of 7^36? (a) 3 (b) 7 (c) 9 (d) 1
7. 7.What is the smallest number you must add to 226 to make it divisible by 3? (a) 2 (b) 1
8. 8.What is the smallest number you must add to 292 to make it divisible by 7? (a) 1 (b) 2 (c) 5 (d) 3
9. 9.What is the remainder when 73 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 2
10. 10.What is the units digit of 4^31? (a) 6 (b) 4 (c) 3 (d) 5
11. 11.What is the smallest number you must add to 410 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
12. 12.Which of these numbers is divisible by 3? (a) 1567 (b) 1569 (c) 1573 (d) 1571
13. 13.What is the remainder when 518 is divided by 12? (a) 3 (b) 2 (c) 10 (d) 1
14. 14.Which of these numbers is divisible by 3? (a) 759 (b) 758 (c) 763 (d) 761
15. 15.Which of these numbers is divisible by 6? (a) 303 (b) 296 (c) 300 (d) 308
16. 16.What is the smallest number you must add to 271 to make it divisible by 12? (a) 4 (b) 5 (c) 7 (d) 6
17. 17.Which of these numbers is divisible by 3? (a) 2419 (b) 2426 (c) 2423 (d) 2421
18. 18.Which of these numbers is divisible by 4? (a) 1548 (b) 1554 (c) 1547 (d) 1551
19. 19.What is the units digit of 3^57? (a) 3 (b) 9 (c) 7 (d) 1
20. 20.Which of these numbers is divisible by 4? (a) 2737 (b) 2735 (c) 2741 (d) 2736
21. 21.What is the smallest number you must add to 202 to make it divisible by 3? (a) 2 (b) 1
22. 22.What is the units digit of 8^44? (a) 6 (b) 4 (c) 8 (d) 2
23. 23.What is the smallest number you must add to 134 to make it divisible by 3? (a) 2 (b) 1
24. 24.What is the remainder when 64 is divided by 3? (a) 1 (b) 0 (c) 2
25. 25.What is the units digit of 2^53? (a) 8 (b) 2 (c) 4 (d) 6
26. 26.What is the smallest number you must add to 511 to make it divisible by 12? (a) 4 (b) 7 (c) 6 (d) 5
27. 27.What is the units digit of 8^24? (a) 2 (b) 4 (c) 8 (d) 6
28. 28.What is the units digit of 8^52? (a) 2 (b) 8 (c) 6 (d) 4
29. 29.What is the units digit of 2^21? (a) 4 (b) 8 (c) 6 (d) 2
30. 30.What is the remainder when 204 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
31. 31.What is the units digit of 4^41? (a) 4 (b) 5 (c) 6 (d) 3
32. 32.Which of these numbers is divisible by 11? (a) 1447 (b) 1460 (c) 1441 (d) 1439
33. 33.Which of these numbers is divisible by 3? (a) 175 (b) 181 (c) 179 (d) 177
34. 34.Which of these numbers is divisible by 3? (a) 1816 (b) 1811 (c) 1813 (d) 1812
35. 35.What is the remainder when 483 is divided by 10? (a) 2 (b) 7 (c) 4 (d) 3
36. 36.What is the remainder when 344 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 1
37. 37.What is the remainder when 86 is divided by 7? (a) 2 (b) 5 (c) 1 (d) 3
38. 38.Which of these numbers is divisible by 4? (a) 3351 (b) 3347 (c) 3348 (d) 3354
39. 39.What is the smallest number you must add to 365 to make it divisible by 7? (a) 5 (b) 4 (c) 1 (d) 6
40. 40.Which of these numbers is divisible by 8? (a) 1095 (b) 1103 (c) 1085 (d) 1088

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