Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 472 · 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 remainder when 162 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
2. 2.What is the remainder when 410 is divided by 7? (a) 3 (b) 5 (c) 6 (d) 4
3. 3.What is the units digit of 8^30? (a) 2 (b) 4 (c) 8 (d) 6
4. 4.What is the remainder when 291 is divided by 8? (a) 2 (b) 4 (c) 3 (d) 5
5. 5.Which of these numbers is divisible by 4? (a) 1966 (b) 1974 (c) 1968 (d) 1971
6. 6.What is the remainder when 648 is divided by 11? (a) 9 (b) 1 (c) 10 (d) 8
7. 7.Which of these numbers is divisible by 11? (a) 2029 (b) 2035 (c) 2049 (d) 2038
8. 8.What is the units digit of 7^18? (a) 7 (b) 1 (c) 3 (d) 9
9. 9.What is the units digit of 3^45? (a) 9 (b) 1 (c) 7 (d) 3
10. 10.What is the units digit of 7^20? (a) 3 (b) 1 (c) 7 (d) 9
11. 11.Which of these numbers is divisible by 9? (a) 5344 (b) 5358 (c) 5352 (d) 5346
12. 12.Which of these numbers is divisible by 9? (a) 2274 (b) 2257 (c) 2259 (d) 2260
13. 13.What is the smallest number you must add to 672 to make it divisible by 10? (a) 2 (b) 7 (c) 9 (d) 8
14. 14.Which of these numbers is divisible by 9? (a) 1046 (b) 1061 (c) 1044 (d) 1036
15. 15.What is the units digit of 8^35? (a) 4 (b) 6 (c) 2 (d) 8
16. 16.What is the units digit of 2^53? (a) 2 (b) 8 (c) 4 (d) 6
17. 17.What is the smallest number you must add to 73 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 3
18. 18.Which of these numbers is divisible by 3? (a) 1504 (b) 1507 (c) 1506 (d) 1510
19. 19.What is the smallest number you must add to 404 to make it divisible by 7? (a) 5 (b) 1 (c) 3 (d) 2
20. 20.Which of these numbers is divisible by 3? (a) 649 (b) 646 (c) 643 (d) 645
21. 21.What is the units digit of 9^40? (a) 9 (b) 2 (c) 0 (d) 1
22. 22.What is the remainder when 342 is divided by 11? (a) 0 (b) 10 (c) 2 (d) 1
23. 23.Which of these numbers is divisible by 3? (a) 547 (b) 550 (c) 546 (d) 545
24. 24.What is the smallest number you must add to 578 to make it divisible by 12? (a) 9 (b) 2 (c) 11 (d) 10
25. 25.What is the remainder when 37 is divided by 3? (a) 1 (b) 0 (c) 2
26. 26.Which of these numbers is divisible by 4? (a) 2507 (b) 2509 (c) 2503 (d) 2504
27. 27.Which of these numbers is divisible by 3? (a) 191 (b) 187 (c) 186 (d) 184
28. 28.Which of these numbers is divisible by 8? (a) 4700 (b) 4707 (c) 4693 (d) 4696
29. 29.Which of these numbers is divisible by 6? (a) 792 (b) 791 (c) 797 (d) 803
30. 30.Which of these numbers is divisible by 4? (a) 896 (b) 897 (c) 894 (d) 902
31. 31.What is the smallest number you must add to 389 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
32. 32.Which of these numbers is divisible by 11? (a) 7119 (b) 7138 (c) 7143 (d) 7128
33. 33.What is the remainder when 394 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
34. 34.What is the remainder when 353 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 1
35. 35.What is the remainder when 415 is divided by 11? (a) 9 (b) 7 (c) 3 (d) 8
36. 36.What is the units digit of 7^50? (a) 3 (b) 7 (c) 1 (d) 9
37. 37.What is the smallest number you must add to 403 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
38. 38.What is the units digit of 2^42? (a) 8 (b) 6 (c) 2 (d) 4
39. 39.Which of these numbers is divisible by 4? (a) 167 (b) 164 (c) 171 (d) 162
40. 40.What is the units digit of 3^56? (a) 7 (b) 9 (c) 1 (d) 3

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