Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 393 · 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 3? (a) 118 (b) 125 (c) 120 (d) 122
2. 2.What is the remainder when 408 is divided by 10? (a) 8 (b) 2 (c) 7 (d) 9
3. 3.What is the remainder when 93 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
4. 4.Which of these numbers is divisible by 11? (a) 8116 (b) 8139 (c) 8120 (d) 8118
5. 5.What is the remainder when 103 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
6. 6.Which of these numbers is divisible by 3? (a) 593 (b) 586 (c) 588 (d) 590
7. 7.What is the remainder when 111 is divided by 7? (a) 4 (b) 6 (c) 1 (d) 5
8. 8.Which of these numbers is divisible by 4? (a) 1038 (b) 1043 (c) 1045 (d) 1040
9. 9.What is the units digit of 4^40? (a) 7 (b) 5 (c) 4 (d) 6
10. 10.Which of these numbers is divisible by 8? (a) 1679 (b) 1672 (c) 1671 (d) 1684
11. 11.What is the smallest number you must add to 517 to make it divisible by 7? (a) 3 (b) 6 (c) 2 (d) 1
12. 12.What is the remainder when 160 is divided by 3? (a) 2 (b) 1 (c) 0
13. 13.What is the smallest number you must add to 562 to make it divisible by 9? (a) 6 (b) 4 (c) 5 (d) 7
14. 14.Which of these numbers is divisible by 4? (a) 3318 (b) 3313 (c) 3323 (d) 3316
15. 15.What is the remainder when 105 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
16. 16.What is the smallest number you must add to 310 to make it divisible by 4? (a) 1 (b) 2 (c) 3
17. 17.What is the units digit of 4^24? (a) 6 (b) 4 (c) 5 (d) 7
18. 18.What is the smallest number you must add to 100 to make it divisible by 3? (a) 2 (b) 1
19. 19.Which of these numbers is divisible by 8? (a) 1152 (b) 1150 (c) 1156 (d) 1167
20. 20.What is the units digit of 7^46? (a) 1 (b) 9 (c) 7 (d) 3
21. 21.What is the units digit of 4^16? (a) 6 (b) 5 (c) 7 (d) 4
22. 22.Which of these numbers is divisible by 9? (a) 5098 (b) 5104 (c) 5094 (d) 5087
23. 23.What is the smallest number you must add to 783 to make it divisible by 10? (a) 3 (b) 8 (c) 7 (d) 6
24. 24.Which of these numbers is divisible by 8? (a) 6289 (b) 6280 (c) 6282 (d) 6277
25. 25.What is the remainder when 567 is divided by 10? (a) 3 (b) 6 (c) 8 (d) 7
26. 26.What is the remainder when 101 is divided by 9? (a) 1 (b) 3 (c) 2 (d) 7
27. 27.What is the remainder when 406 is divided by 11? (a) 8 (b) 10 (c) 1 (d) 9
28. 28.What is the smallest number you must add to 243 to make it divisible by 12? (a) 8 (b) 10 (c) 3 (d) 9
29. 29.Which of these numbers is divisible by 8? (a) 4798 (b) 4804 (c) 4809 (d) 4800
30. 30.What is the units digit of 4^33? (a) 6 (b) 4 (c) 5 (d) 3
31. 31.Which of these numbers is divisible by 3? (a) 2509 (b) 2513 (c) 2508 (d) 2507
32. 32.Which of these numbers is divisible by 8? (a) 4266 (b) 4257 (c) 4249 (d) 4256
33. 33.What is the smallest number you must add to 690 to make it divisible by 12? (a) 6 (b) 7 (c) 8 (d) 5
34. 34.What is the smallest number you must add to 331 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
35. 35.Which of these numbers is divisible by 6? (a) 4942 (b) 4937 (c) 4938 (d) 4948
36. 36.Which of these numbers is divisible by 3? (a) 2475 (b) 2480 (c) 2476 (d) 2473
37. 37.Which of these numbers is divisible by 9? (a) 5549 (b) 5544 (c) 5539 (d) 5555
38. 38.Which of these numbers is divisible by 6? (a) 4288 (b) 4273 (c) 4278 (d) 4283
39. 39.What is the units digit of 4^49? (a) 5 (b) 4 (c) 3 (d) 6
40. 40.What is the units digit of 8^13? (a) 2 (b) 4 (c) 6 (d) 8

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