Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 621 · 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 535 to make it divisible by 11? (a) 5 (b) 4 (c) 7 (d) 3
2. 2.Which of these numbers is divisible by 9? (a) 3573 (b) 3590 (c) 3572 (d) 3580
3. 3.Which of these numbers is divisible by 4? (a) 3101 (b) 3111 (c) 3107 (d) 3104
4. 4.What is the smallest number you must add to 444 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
5. 5.Which of these numbers is divisible by 9? (a) 856 (b) 880 (c) 868 (d) 864
6. 6.Which of these numbers is divisible by 11? (a) 9273 (b) 9294 (c) 9267 (d) 9274
7. 7.Which of these numbers is divisible by 6? (a) 1194 (b) 1193 (c) 1204 (d) 1197
8. 8.What is the smallest number you must add to 188 to make it divisible by 3? (a) 2 (b) 1
9. 9.Which of these numbers is divisible by 6? (a) 274 (b) 280 (c) 286 (d) 276
10. 10.What is the units digit of 9^37? (a) 7 (b) 8 (c) 1 (d) 9
11. 11.What is the smallest number you must add to 136 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
12. 12.What is the smallest number you must add to 429 to make it divisible by 9? (a) 2 (b) 6 (c) 3 (d) 4
13. 13.Which of these numbers is divisible by 3? (a) 168 (b) 173 (c) 167 (d) 169
14. 14.What is the units digit of 2^53? (a) 6 (b) 8 (c) 2 (d) 4
15. 15.Which of these numbers is divisible by 11? (a) 628 (b) 640 (c) 626 (d) 627
16. 16.What is the smallest number you must add to 310 to make it divisible by 4? (a) 2 (b) 3 (c) 1
17. 17.What is the units digit of 9^54? (a) 0 (b) 9 (c) 2 (d) 1
18. 18.What is the smallest number you must add to 113 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
19. 19.What is the remainder when 312 is divided by 10? (a) 2 (b) 1 (c) 3 (d) 8
20. 20.What is the units digit of 7^12? (a) 3 (b) 1 (c) 9 (d) 7
21. 21.What is the remainder when 191 is divided by 11? (a) 7 (b) 3 (c) 5 (d) 4
22. 22.What is the units digit of 2^20? (a) 8 (b) 4 (c) 6 (d) 2
23. 23.What is the smallest number you must add to 211 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
24. 24.Which of these numbers is divisible by 8? (a) 853 (b) 842 (c) 848 (d) 859
25. 25.What is the units digit of 8^38? (a) 2 (b) 8 (c) 6 (d) 4
26. 26.What is the remainder when 807 is divided by 12? (a) 2 (b) 9 (c) 3 (d) 4
27. 27.What is the units digit of 8^54? (a) 6 (b) 8 (c) 2 (d) 4
28. 28.What is the smallest number you must add to 598 to make it divisible by 10? (a) 8 (b) 1 (c) 2 (d) 3
29. 29.What is the remainder when 484 is divided by 10? (a) 5 (b) 4 (c) 3 (d) 6
30. 30.Which of these numbers is divisible by 3? (a) 880 (b) 874 (c) 877 (d) 876
31. 31.What is the smallest number you must add to 237 to make it divisible by 4? (a) 3 (b) 2 (c) 1
32. 32.Which of these numbers is divisible by 4? (a) 650 (b) 659 (c) 652 (d) 655
33. 33.Which of these numbers is divisible by 8? (a) 320 (b) 335 (c) 316 (d) 323
34. 34.What is the units digit of 2^33? (a) 4 (b) 2 (c) 6 (d) 8
35. 35.What is the smallest number you must add to 402 to make it divisible by 10? (a) 9 (b) 7 (c) 2 (d) 8
36. 36.What is the remainder when 401 is divided by 7? (a) 5 (b) 1 (c) 2 (d) 3
37. 37.What is the units digit of 4^56? (a) 4 (b) 7 (c) 5 (d) 6
38. 38.What is the smallest number you must add to 211 to make it divisible by 11? (a) 2 (b) 8 (c) 10 (d) 9
39. 39.What is the smallest number you must add to 157 to make it divisible by 4? (a) 1 (b) 2 (c) 3
40. 40.What is the remainder when 241 is divided by 3? (a) 0 (b) 2 (c) 1

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