Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 551 · 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 6? (a) 907 (b) 899 (c) 903 (d) 900
2. 2.What is the smallest number you must add to 302 to make it divisible by 10? (a) 9 (b) 2 (c) 7 (d) 8
3. 3.What is the units digit of 4^59? (a) 5 (b) 4 (c) 6 (d) 3
4. 4.What is the remainder when 75 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
5. 5.What is the remainder when 455 is divided by 8? (a) 7 (b) 1 (c) 5 (d) 6
6. 6.What is the smallest number you must add to 92 to make it divisible by 3? (a) 1 (b) 2
7. 7.What is the units digit of 2^15? (a) 2 (b) 6 (c) 8 (d) 4
8. 8.What is the remainder when 458 is divided by 8? (a) 3 (b) 6 (c) 1 (d) 2
9. 9.What is the smallest number you must add to 126 to make it divisible by 8? (a) 6 (b) 3 (c) 2 (d) 1
10. 10.What is the units digit of 3^20? (a) 7 (b) 3 (c) 1 (d) 9
11. 11.Which of these numbers is divisible by 4? (a) 1569 (b) 1564 (c) 1563 (d) 1567
12. 12.Which of these numbers is divisible by 11? (a) 2764 (b) 2772 (c) 2776 (d) 2793
13. 13.What is the remainder when 94 is divided by 3? (a) 2 (b) 0 (c) 1
14. 14.What is the smallest number you must add to 59 to make it divisible by 3? (a) 1 (b) 2
15. 15.What is the smallest number you must add to 236 to make it divisible by 3? (a) 1 (b) 2
16. 16.Which of these numbers is divisible by 11? (a) 9235 (b) 9219 (c) 9217 (d) 9218
17. 17.What is the units digit of 3^25? (a) 1 (b) 3 (c) 9 (d) 7
18. 18.What is the remainder when 428 is divided by 12? (a) 7 (b) 8 (c) 9 (d) 4
19. 19.What is the remainder when 314 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
20. 20.Which of these numbers is divisible by 4? (a) 1027 (b) 1018 (c) 1020 (d) 1023
21. 21.What is the smallest number you must add to 140 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
22. 22.What is the smallest number you must add to 517 to make it divisible by 10? (a) 2 (b) 7 (c) 4 (d) 3
23. 23.What is the units digit of 8^58? (a) 8 (b) 2 (c) 6 (d) 4
24. 24.Which of these numbers is divisible by 9? (a) 3176 (b) 3165 (c) 3183 (d) 3168
25. 25.Which of these numbers is divisible by 11? (a) 2551 (b) 2541 (c) 2539 (d) 2560
26. 26.Which of these numbers is divisible by 9? (a) 7181 (b) 7164 (c) 7161 (d) 7169
27. 27.Which of these numbers is divisible by 8? (a) 2865 (b) 2878 (c) 2864 (d) 2857
28. 28.What is the units digit of 3^30? (a) 9 (b) 3 (c) 1 (d) 7
29. 29.Which of these numbers is divisible by 9? (a) 3766 (b) 3781 (c) 3774 (d) 3771
30. 30.What is the units digit of 7^12? (a) 1 (b) 9 (c) 3 (d) 7
31. 31.What is the units digit of 9^46? (a) 2 (b) 9 (c) 0 (d) 1
32. 32.What is the smallest number you must add to 144 to make it divisible by 11? (a) 10 (b) 9 (c) 8 (d) 1
33. 33.What is the smallest number you must add to 132 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
34. 34.What is the smallest number you must add to 173 to make it divisible by 7? (a) 5 (b) 3 (c) 2 (d) 1
35. 35.Which of these numbers is divisible by 8? (a) 3360 (b) 3355 (c) 3371 (d) 3367
36. 36.What is the remainder when 161 is divided by 3? (a) 0 (b) 1 (c) 2
37. 37.What is the smallest number you must add to 602 to make it divisible by 8? (a) 5 (b) 2 (c) 7 (d) 6
38. 38.What is the units digit of 7^24? (a) 9 (b) 7 (c) 3 (d) 1
39. 39.Which of these numbers is divisible by 3? (a) 571 (b) 575 (c) 573 (d) 578
40. 40.What is the units digit of 2^17? (a) 8 (b) 6 (c) 4 (d) 2

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