Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 396 · 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 4^48? (a) 7 (b) 5 (c) 4 (d) 6
2. 2.What is the units digit of 3^56? (a) 7 (b) 9 (c) 1 (d) 3
3. 3.What is the smallest number you must add to 751 to make it divisible by 12? (a) 6 (b) 7 (c) 5 (d) 4
4. 4.What is the units digit of 7^56? (a) 1 (b) 7 (c) 3 (d) 9
5. 5.Which of these numbers is divisible by 8? (a) 4550 (b) 4566 (c) 4552 (d) 4555
6. 6.What is the units digit of 3^45? (a) 3 (b) 9 (c) 1 (d) 7
7. 7.What is the remainder when 375 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 7
8. 8.What is the smallest number you must add to 63 to make it divisible by 4? (a) 1 (b) 2 (c) 3
9. 9.Which of these numbers is divisible by 9? (a) 2844 (b) 2859 (c) 2838 (d) 2845
10. 10.What is the units digit of 4^13? (a) 4 (b) 3 (c) 6 (d) 5
11. 11.Which of these numbers is divisible by 3? (a) 420 (b) 419 (c) 422 (d) 424
12. 12.What is the units digit of 9^20? (a) 9 (b) 0 (c) 2 (d) 1
13. 13.What is the smallest number you must add to 42 to make it divisible by 4? (a) 1 (b) 3 (c) 2
14. 14.What is the units digit of 8^26? (a) 4 (b) 8 (c) 2 (d) 6
15. 15.Which of these numbers is divisible by 8? (a) 1875 (b) 1870 (c) 1872 (d) 1886
16. 16.Which of these numbers is divisible by 3? (a) 1295 (b) 1297 (c) 1291 (d) 1293
17. 17.What is the smallest number you must add to 94 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
18. 18.What is the remainder when 257 is divided by 12? (a) 4 (b) 6 (c) 5 (d) 7
19. 19.Which of these numbers is divisible by 3? (a) 821 (b) 817 (c) 815 (d) 816
20. 20.What is the smallest number you must add to 139 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
21. 21.What is the units digit of 4^49? (a) 6 (b) 5 (c) 3 (d) 4
22. 22.What is the smallest number you must add to 387 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
23. 23.What is the units digit of 7^35? (a) 7 (b) 3 (c) 9 (d) 1
24. 24.What is the remainder when 196 is divided by 3? (a) 1 (b) 0 (c) 2
25. 25.What is the smallest number you must add to 286 to make it divisible by 9? (a) 3 (b) 7 (c) 2 (d) 1
26. 26.Which of these numbers is divisible by 11? (a) 2476 (b) 2500 (c) 2496 (d) 2486
27. 27.What is the units digit of 2^55? (a) 4 (b) 6 (c) 2 (d) 8
28. 28.Which of these numbers is divisible by 8? (a) 2050 (b) 2071 (c) 2056 (d) 2062
29. 29.What is the smallest number you must add to 388 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
30. 30.What is the smallest number you must add to 318 to make it divisible by 10? (a) 3 (b) 8 (c) 1 (d) 2
31. 31.Which of these numbers is divisible by 6? (a) 754 (b) 749 (c) 758 (d) 750
32. 32.What is the remainder when 426 is divided by 7? (a) 6 (b) 5 (c) 1 (d) 4
33. 33.What is the smallest number you must add to 571 to make it divisible by 10? (a) 9 (b) 1 (c) 7 (d) 8
34. 34.What is the units digit of 9^14? (a) 2 (b) 9 (c) 1 (d) 0
35. 35.What is the smallest number you must add to 79 to make it divisible by 4? (a) 2 (b) 1 (c) 3
36. 36.Which of these numbers is divisible by 11? (a) 7075 (b) 7070 (c) 7062 (d) 7059
37. 37.What is the remainder when 178 is divided by 3? (a) 2 (b) 0 (c) 1
38. 38.Which of these numbers is divisible by 6? (a) 4352 (b) 4358 (c) 4350 (d) 4346
39. 39.What is the units digit of 8^24? (a) 2 (b) 6 (c) 8 (d) 4
40. 40.Which of these numbers is divisible by 11? (a) 1773 (b) 1759 (c) 1760 (d) 1764

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