Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 992 · 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 11? (a) 8176 (b) 8202 (c) 8184 (d) 8194
2. 2.What is the remainder when 205 is divided by 7? (a) 3 (b) 5 (c) 2 (d) 1
3. 3.What is the remainder when 351 is divided by 10? (a) 2 (b) 0 (c) 1 (d) 9
4. 4.What is the smallest number you must add to 146 to make it divisible by 3? (a) 2 (b) 1
5. 5.Which of these numbers is divisible by 6? (a) 3613 (b) 3623 (c) 3618 (d) 3626
6. 6.What is the remainder when 463 is divided by 11? (a) 0 (b) 2 (c) 1 (d) 10
7. 7.What is the smallest number you must add to 170 to make it divisible by 3? (a) 1 (b) 2
8. 8.What is the units digit of 2^56? (a) 4 (b) 6 (c) 2 (d) 8
9. 9.What is the remainder when 683 is divided by 11? (a) 0 (b) 1 (c) 2 (d) 10
10. 10.What is the smallest number you must add to 73 to make it divisible by 3? (a) 1 (b) 2
11. 11.Which of these numbers is divisible by 4? (a) 2931 (b) 2933 (c) 2932 (d) 2938
12. 12.Which of these numbers is divisible by 6? (a) 2324 (b) 2317 (c) 2316 (d) 2315
13. 13.What is the units digit of 7^34? (a) 7 (b) 3 (c) 1 (d) 9
14. 14.What is the units digit of 4^41? (a) 6 (b) 3 (c) 4 (d) 5
15. 15.What is the units digit of 9^13? (a) 1 (b) 7 (c) 9 (d) 8
16. 16.What is the remainder when 133 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
17. 17.Which of these numbers is divisible by 4? (a) 289 (b) 284 (c) 281 (d) 287
18. 18.Which of these numbers is divisible by 11? (a) 6825 (b) 6818 (c) 6836 (d) 6820
19. 19.Which of these numbers is divisible by 6? (a) 4326 (b) 4324 (c) 4327 (d) 4335
20. 20.What is the units digit of 9^17? (a) 9 (b) 8 (c) 1 (d) 7
21. 21.What is the remainder when 478 is divided by 9? (a) 0 (b) 8 (c) 1 (d) 2
22. 22.What is the units digit of 2^57? (a) 6 (b) 4 (c) 8 (d) 2
23. 23.Which of these numbers is divisible by 6? (a) 2418 (b) 2420 (c) 2417 (d) 2425
24. 24.What is the smallest number you must add to 144 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 7
25. 25.What is the units digit of 7^13? (a) 7 (b) 9 (c) 3 (d) 1
26. 26.What is the smallest number you must add to 269 to make it divisible by 9? (a) 3 (b) 1 (c) 8 (d) 2
27. 27.What is the smallest number you must add to 173 to make it divisible by 3? (a) 2 (b) 1
28. 28.What is the units digit of 8^29? (a) 6 (b) 2 (c) 4 (d) 8
29. 29.What is the smallest number you must add to 148 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 4
30. 30.What is the units digit of 8^40? (a) 4 (b) 8 (c) 2 (d) 6
31. 31.What is the units digit of 4^45? (a) 5 (b) 6 (c) 3 (d) 4
32. 32.What is the remainder when 340 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
33. 33.What is the units digit of 7^22? (a) 7 (b) 1 (c) 9 (d) 3
34. 34.Which of these numbers is divisible by 3? (a) 2698 (b) 2700 (c) 2705 (d) 2701
35. 35.What is the units digit of 3^32? (a) 7 (b) 9 (c) 1 (d) 3
36. 36.What is the smallest number you must add to 403 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
37. 37.What is the units digit of 3^37? (a) 9 (b) 7 (c) 1 (d) 3
38. 38.What is the smallest number you must add to 236 to make it divisible by 10? (a) 3 (b) 5 (c) 6 (d) 4
39. 39.Which of these numbers is divisible by 11? (a) 3153 (b) 3136 (c) 3127 (d) 3135
40. 40.What is the remainder when 807 is divided by 12? (a) 9 (b) 4 (c) 2 (d) 3

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