Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 994 · 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 units digit of 7^16? (a) 3 (b) 9 (c) 1 (d) 7
2. 2.Which of these numbers is divisible by 9? (a) 4244 (b) 4235 (c) 4253 (d) 4239
3. 3.Which of these numbers is divisible by 9? (a) 3610 (b) 3600 (c) 3593 (d) 3602
4. 4.Which of these numbers is divisible by 6? (a) 1115 (b) 1102 (c) 1104 (d) 1105
5. 5.What is the smallest number you must add to 184 to make it divisible by 3? (a) 1 (b) 2
6. 6.What is the remainder when 261 is divided by 11? (a) 9 (b) 8 (c) 7 (d) 3
7. 7.What is the smallest number you must add to 348 to make it divisible by 11? (a) 5 (b) 3 (c) 7 (d) 4
8. 8.Which of these numbers is divisible by 3? (a) 1660 (b) 1663 (c) 1657 (d) 1659
9. 9.What is the units digit of 8^14? (a) 8 (b) 2 (c) 4 (d) 6
10. 10.What is the remainder when 158 is divided by 10? (a) 2 (b) 7 (c) 8 (d) 9
11. 11.What is the remainder when 46 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
12. 12.What is the units digit of 7^31? (a) 7 (b) 1 (c) 3 (d) 9
13. 13.What is the remainder when 79 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
14. 14.What is the remainder when 195 is divided by 12? (a) 9 (b) 3 (c) 2 (d) 4
15. 15.What is the smallest number you must add to 556 to make it divisible by 9? (a) 2 (b) 7 (c) 3 (d) 1
16. 16.What is the units digit of 9^40? (a) 0 (b) 2 (c) 1 (d) 9
17. 17.Which of these numbers is divisible by 4? (a) 1286 (b) 1279 (c) 1282 (d) 1280
18. 18.What is the smallest number you must add to 146 to make it divisible by 3? (a) 1 (b) 2
19. 19.Which of these numbers is divisible by 8? (a) 4548 (b) 4556 (c) 4538 (d) 4544
20. 20.What is the smallest number you must add to 288 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
21. 21.What is the units digit of 2^39? (a) 6 (b) 8 (c) 2 (d) 4
22. 22.What is the smallest number you must add to 37 to make it divisible by 3? (a) 2 (b) 1
23. 23.What is the units digit of 9^56? (a) 9 (b) 1 (c) 0 (d) 2
24. 24.Which of these numbers is divisible by 11? (a) 8779 (b) 8778 (c) 8768 (d) 8795
25. 25.What is the smallest number you must add to 202 to make it divisible by 10? (a) 7 (b) 2 (c) 8 (d) 9
26. 26.What is the units digit of 3^32? (a) 3 (b) 9 (c) 7 (d) 1
27. 27.What is the smallest number you must add to 209 to make it divisible by 4? (a) 2 (b) 3 (c) 1
28. 28.What is the units digit of 8^46? (a) 6 (b) 4 (c) 8 (d) 2
29. 29.Which of these numbers is divisible by 6? (a) 842 (b) 846 (c) 853 (d) 847
30. 30.What is the units digit of 3^50? (a) 9 (b) 1 (c) 3 (d) 7
31. 31.What is the remainder when 116 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 0
32. 32.What is the units digit of 3^17? (a) 3 (b) 1 (c) 7 (d) 9
33. 33.What is the units digit of 8^19? (a) 2 (b) 8 (c) 6 (d) 4
34. 34.What is the remainder when 282 is divided by 10? (a) 1 (b) 8 (c) 2 (d) 3
35. 35.Which of these numbers is divisible by 8? (a) 3010 (b) 3017 (c) 3031 (d) 3016
36. 36.What is the units digit of 9^35? (a) 7 (b) 1 (c) 9 (d) 8
37. 37.What is the smallest number you must add to 170 to make it divisible by 11? (a) 5 (b) 7 (c) 6 (d) 8
38. 38.What is the units digit of 7^19? (a) 7 (b) 1 (c) 3 (d) 9
39. 39.Which of these numbers is divisible by 9? (a) 6362 (b) 6340 (c) 6345 (d) 6346
40. 40.What is the smallest number you must add to 129 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2

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