Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 55 · 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.Which of these numbers is divisible by 9? (a) 7076 (b) 7084 (c) 7067 (d) 7074
2. 2.What is the units digit of 8^31? (a) 8 (b) 2 (c) 4 (d) 6
3. 3.What is the smallest number you must add to 631 to make it divisible by 9? (a) 1 (b) 7 (c) 6 (d) 8
4. 4.What is the units digit of 7^41? (a) 1 (b) 3 (c) 9 (d) 7
5. 5.Which of these numbers is divisible by 8? (a) 6054 (b) 6056 (c) 6068 (d) 6063
6. 6.What is the units digit of 7^11? (a) 7 (b) 1 (c) 9 (d) 3
7. 7.What is the smallest number you must add to 530 to make it divisible by 12? (a) 11 (b) 9 (c) 2 (d) 10
8. 8.What is the smallest number you must add to 138 to make it divisible by 9? (a) 7 (b) 5 (c) 3 (d) 6
9. 9.Which of these numbers is divisible by 4? (a) 3030 (b) 3035 (c) 3026 (d) 3028
10. 10.What is the remainder when 166 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
11. 11.What is the remainder when 242 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
12. 12.What is the units digit of 4^47? (a) 3 (b) 6 (c) 4 (d) 5
13. 13.What is the remainder when 270 is divided by 8? (a) 5 (b) 7 (c) 2 (d) 6
14. 14.What is the remainder when 74 is divided by 3? (a) 0 (b) 2 (c) 1
15. 15.Which of these numbers is divisible by 3? (a) 2694 (b) 2692 (c) 2699 (d) 2696
16. 16.What is the smallest number you must add to 718 to make it divisible by 9? (a) 2 (b) 7 (c) 1 (d) 3
17. 17.What is the units digit of 2^19? (a) 2 (b) 6 (c) 4 (d) 8
18. 18.Which of these numbers is divisible by 3? (a) 1672 (b) 1671 (c) 1676 (d) 1669
19. 19.What is the units digit of 9^45? (a) 1 (b) 7 (c) 9 (d) 8
20. 20.What is the remainder when 278 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
21. 21.What is the smallest number you must add to 372 to make it divisible by 10? (a) 2 (b) 7 (c) 8 (d) 9
22. 22.What is the smallest number you must add to 373 to make it divisible by 10? (a) 8 (b) 3 (c) 6 (d) 7
23. 23.Which of these numbers is divisible by 3? (a) 605 (b) 607 (c) 601 (d) 603
24. 24.What is the units digit of 9^27? (a) 1 (b) 8 (c) 7 (d) 9
25. 25.What is the remainder when 549 is divided by 7? (a) 3 (b) 2 (c) 4 (d) 5
26. 26.Which of these numbers is divisible by 11? (a) 4263 (b) 4256 (c) 4257 (d) 4273
27. 27.Which of these numbers is divisible by 4? (a) 2841 (b) 2836 (c) 2834 (d) 2838
28. 28.What is the units digit of 3^59? (a) 3 (b) 1 (c) 9 (d) 7
29. 29.What is the units digit of 7^59? (a) 3 (b) 1 (c) 7 (d) 9
30. 30.What is the remainder when 89 is divided by 6? (a) 5 (b) 3 (c) 1 (d) 4
31. 31.What is the smallest number you must add to 696 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 7
32. 32.What is the smallest number you must add to 174 to make it divisible by 12? (a) 6 (b) 5 (c) 7 (d) 8
33. 33.What is the smallest number you must add to 588 to make it divisible by 11? (a) 5 (b) 7 (c) 6 (d) 8
34. 34.What is the units digit of 9^57? (a) 7 (b) 1 (c) 9 (d) 8
35. 35.Which of these numbers is divisible by 9? (a) 3168 (b) 3184 (c) 3175 (d) 3163
36. 36.What is the remainder when 229 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
37. 37.Which of these numbers is divisible by 8? (a) 4996 (b) 5009 (c) 5000 (d) 5006
38. 38.What is the smallest number you must add to 94 to make it divisible by 7? (a) 6 (b) 3 (c) 4 (d) 5
39. 39.Which of these numbers is divisible by 9? (a) 6047 (b) 6048 (c) 6058 (d) 6056
40. 40.Which of these numbers is divisible by 8? (a) 2385 (b) 2392 (c) 2404 (d) 2397

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