Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 937 · 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 91 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
2. 2.Which of these numbers is divisible by 9? (a) 7391 (b) 7406 (c) 7415 (d) 7398
3. 3.What is the smallest number you must add to 79 to make it divisible by 4? (a) 2 (b) 1 (c) 3
4. 4.Which of these numbers is divisible by 6? (a) 2064 (b) 2062 (c) 2074 (d) 2065
5. 5.What is the remainder when 391 is divided by 10? (a) 0 (b) 2 (c) 9 (d) 1
6. 6.What is the remainder when 377 is divided by 11? (a) 2 (b) 4 (c) 3 (d) 8
7. 7.What is the remainder when 457 is divided by 6? (a) 5 (b) 2 (c) 0 (d) 1
8. 8.What is the smallest number you must add to 961 to make it divisible by 12? (a) 10 (b) 1 (c) 9 (d) 11
9. 9.Which of these numbers is divisible by 6? (a) 5391 (b) 5382 (c) 5377 (d) 5383
10. 10.What is the remainder when 152 is divided by 11? (a) 9 (b) 2 (c) 10 (d) 8
11. 11.Which of these numbers is divisible by 4? (a) 2450 (b) 2442 (c) 2444 (d) 2446
12. 12.What is the units digit of 2^37? (a) 8 (b) 4 (c) 6 (d) 2
13. 13.What is the units digit of 7^35? (a) 9 (b) 1 (c) 7 (d) 3
14. 14.What is the remainder when 480 is divided by 9? (a) 3 (b) 6 (c) 2 (d) 4
15. 15.What is the units digit of 4^35? (a) 6 (b) 4 (c) 5 (d) 3
16. 16.What is the smallest number you must add to 398 to make it divisible by 7? (a) 2 (b) 1 (c) 6 (d) 3
17. 17.What is the smallest number you must add to 101 to make it divisible by 3? (a) 2 (b) 1
18. 18.Which of these numbers is divisible by 11? (a) 5467 (b) 5462 (c) 5488 (d) 5468
19. 19.What is the units digit of 4^34? (a) 4 (b) 5 (c) 6 (d) 7
20. 20.What is the remainder when 121 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
21. 21.What is the smallest number you must add to 338 to make it divisible by 12? (a) 10 (b) 11 (c) 2 (d) 9
22. 22.What is the units digit of 2^14? (a) 6 (b) 8 (c) 4 (d) 2
23. 23.What is the units digit of 3^20? (a) 7 (b) 9 (c) 1 (d) 3
24. 24.Which of these numbers is divisible by 8? (a) 2849 (b) 2848 (c) 2860 (d) 2845
25. 25.Which of these numbers is divisible by 4? (a) 2030 (b) 2026 (c) 2033 (d) 2028
26. 26.What is the remainder when 186 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
27. 27.What is the remainder when 118 is divided by 3? (a) 1 (b) 2 (c) 0
28. 28.Which of these numbers is divisible by 6? (a) 4260 (b) 4255 (c) 4267 (d) 4262
29. 29.What is the units digit of 2^24? (a) 8 (b) 6 (c) 4 (d) 2
30. 30.Which of these numbers is divisible by 11? (a) 8848 (b) 8828 (c) 8838 (d) 8833
31. 31.What is the remainder when 118 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
32. 32.What is the remainder when 124 is divided by 3? (a) 2 (b) 1 (c) 0
33. 33.Which of these numbers is divisible by 11? (a) 2549 (b) 2524 (c) 2530 (d) 2536
34. 34.What is the remainder when 271 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
35. 35.Which of these numbers is divisible by 3? (a) 700 (b) 699 (c) 698 (d) 703
36. 36.What is the units digit of 2^55? (a) 2 (b) 8 (c) 4 (d) 6
37. 37.What is the remainder when 462 is divided by 9? (a) 2 (b) 3 (c) 6 (d) 4
38. 38.What is the smallest number you must add to 382 to make it divisible by 7? (a) 5 (b) 2 (c) 3 (d) 4
39. 39.Which of these numbers is divisible by 6? (a) 3115 (b) 3109 (c) 3114 (d) 3121
40. 40.What is the units digit of 2^52? (a) 2 (b) 8 (c) 6 (d) 4

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