Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 415 · 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 smallest number you must add to 533 to make it divisible by 7? (a) 6 (b) 5 (c) 1 (d) 4
2. 2.What is the remainder when 178 is divided by 11? (a) 9 (b) 2 (c) 3 (d) 1
3. 3.What is the smallest number you must add to 437 to make it divisible by 6? (a) 2 (b) 5 (c) 1 (d) 3
4. 4.What is the units digit of 8^41? (a) 8 (b) 2 (c) 4 (d) 6
5. 5.What is the units digit of 2^28? (a) 6 (b) 8 (c) 2 (d) 4
6. 6.What is the smallest number you must add to 345 to make it divisible by 11? (a) 7 (b) 4 (c) 8 (d) 6
7. 7.Which of these numbers is divisible by 3? (a) 1483 (b) 1477 (c) 1481 (d) 1479
8. 8.Which of these numbers is divisible by 4? (a) 279 (b) 275 (c) 276 (d) 281
9. 9.What is the remainder when 185 is divided by 8? (a) 7 (b) 1 (c) 0 (d) 2
10. 10.What is the remainder when 109 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
11. 11.What is the units digit of 2^48? (a) 8 (b) 6 (c) 2 (d) 4
12. 12.What is the remainder when 160 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
13. 13.Which of these numbers is divisible by 6? (a) 3972 (b) 3976 (c) 3967 (d) 3980
14. 14.Which of these numbers is divisible by 6? (a) 1748 (b) 1740 (c) 1741 (d) 1735
15. 15.What is the smallest number you must add to 351 to make it divisible by 10? (a) 9 (b) 8 (c) 7 (d) 1
16. 16.What is the remainder when 128 is divided by 11? (a) 8 (b) 4 (c) 7 (d) 6
17. 17.What is the remainder when 411 is divided by 9? (a) 7 (b) 5 (c) 3 (d) 6
18. 18.Which of these numbers is divisible by 6? (a) 4413 (b) 4400 (c) 4404 (d) 4407
19. 19.What is the smallest number you must add to 275 to make it divisible by 4? (a) 3 (b) 1 (c) 2
20. 20.What is the smallest number you must add to 467 to make it divisible by 10? (a) 7 (b) 2 (c) 4 (d) 3
21. 21.Which of these numbers is divisible by 4? (a) 382 (b) 391 (c) 384 (d) 386
22. 22.What is the remainder when 448 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
23. 23.What is the smallest number you must add to 92 to make it divisible by 3? (a) 1 (b) 2
24. 24.What is the units digit of 7^33? (a) 3 (b) 7 (c) 1 (d) 9
25. 25.What is the units digit of 3^41? (a) 7 (b) 1 (c) 3 (d) 9
26. 26.Which of these numbers is divisible by 6? (a) 433 (b) 428 (c) 441 (d) 432
27. 27.What is the units digit of 8^59? (a) 6 (b) 8 (c) 2 (d) 4
28. 28.What is the smallest number you must add to 567 to make it divisible by 12? (a) 3 (b) 8 (c) 10 (d) 9
29. 29.What is the remainder when 652 is divided by 12? (a) 8 (b) 3 (c) 4 (d) 5
30. 30.Which of these numbers is divisible by 9? (a) 4412 (b) 4399 (c) 4401 (d) 4405
31. 31.Which of these numbers is divisible by 3? (a) 2639 (b) 2633 (c) 2635 (d) 2634
32. 32.Which of these numbers is divisible by 11? (a) 8047 (b) 8030 (c) 8027 (d) 8036
33. 33.What is the units digit of 8^13? (a) 8 (b) 6 (c) 4 (d) 2
34. 34.Which of these numbers is divisible by 3? (a) 2388 (b) 2390 (c) 2393 (d) 2386
35. 35.Which of these numbers is divisible by 4? (a) 1489 (b) 1487 (c) 1488 (d) 1493
36. 36.Which of these numbers is divisible by 3? (a) 220 (b) 217 (c) 216 (d) 215
37. 37.What is the units digit of 7^21? (a) 3 (b) 9 (c) 1 (d) 7
38. 38.What is the smallest number you must add to 523 to make it divisible by 11? (a) 7 (b) 4 (c) 5 (d) 6
39. 39.Which of these numbers is divisible by 9? (a) 4803 (b) 4780 (c) 4796 (d) 4788
40. 40.What is the smallest number you must add to 126 to make it divisible by 8? (a) 3 (b) 2 (c) 6 (d) 1

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