Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 798 · 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 138 to make it divisible by 12? (a) 8 (b) 7 (c) 6 (d) 5
2. 2.Which of these numbers is divisible by 3? (a) 2368 (b) 2371 (c) 2366 (d) 2367
3. 3.What is the smallest number you must add to 877 to make it divisible by 12? (a) 9 (b) 1 (c) 11 (d) 10
4. 4.Which of these numbers is divisible by 3? (a) 2419 (b) 2422 (c) 2416 (d) 2418
5. 5.What is the smallest number you must add to 154 to make it divisible by 3? (a) 1 (b) 2
6. 6.What is the smallest number you must add to 242 to make it divisible by 4? (a) 3 (b) 1 (c) 2
7. 7.What is the units digit of 8^32? (a) 6 (b) 8 (c) 2 (d) 4
8. 8.What is the units digit of 2^26? (a) 6 (b) 2 (c) 4 (d) 8
9. 9.What is the smallest number you must add to 108 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
10. 10.What is the units digit of 3^37? (a) 9 (b) 3 (c) 1 (d) 7
11. 11.Which of these numbers is divisible by 6? (a) 2954 (b) 2969 (c) 2958 (d) 2962
12. 12.What is the remainder when 483 is divided by 8? (a) 5 (b) 4 (c) 2 (d) 3
13. 13.Which of these numbers is divisible by 8? (a) 3345 (b) 3352 (c) 3356 (d) 3362
14. 14.What is the remainder when 788 is divided by 10? (a) 7 (b) 8 (c) 2 (d) 9
15. 15.Which of these numbers is divisible by 11? (a) 2902 (b) 2918 (c) 2913 (d) 2904
16. 16.What is the units digit of 8^56? (a) 8 (b) 2 (c) 4 (d) 6
17. 17.What is the remainder when 458 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 1
18. 18.Which of these numbers is divisible by 4? (a) 3110 (b) 3102 (c) 3104 (d) 3107
19. 19.Which of these numbers is divisible by 11? (a) 3816 (b) 3805 (c) 3806 (d) 3822
20. 20.Which of these numbers is divisible by 4? (a) 1432 (b) 1433 (c) 1431 (d) 1439
21. 21.What is the units digit of 7^41? (a) 1 (b) 9 (c) 3 (d) 7
22. 22.What is the smallest number you must add to 89 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
23. 23.What is the smallest number you must add to 82 to make it divisible by 3? (a) 1 (b) 2
24. 24.What is the remainder when 89 is divided by 3? (a) 2 (b) 1 (c) 0
25. 25.What is the remainder when 859 is divided by 11? (a) 0 (b) 2 (c) 1 (d) 10
26. 26.What is the smallest number you must add to 349 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
27. 27.Which of these numbers is divisible by 9? (a) 7815 (b) 7835 (c) 7821 (d) 7827
28. 28.What is the units digit of 2^18? (a) 6 (b) 8 (c) 2 (d) 4
29. 29.What is the remainder when 94 is divided by 8? (a) 7 (b) 5 (c) 2 (d) 6
30. 30.What is the remainder when 590 is divided by 12? (a) 2 (b) 3 (c) 10 (d) 1
31. 31.What is the units digit of 2^49? (a) 4 (b) 8 (c) 2 (d) 6
32. 32.What is the remainder when 246 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
33. 33.Which of these numbers is divisible by 4? (a) 1146 (b) 1148 (c) 1154 (d) 1149
34. 34.What is the units digit of 9^13? (a) 8 (b) 7 (c) 1 (d) 9
35. 35.What is the smallest number you must add to 316 to make it divisible by 11? (a) 8 (b) 3 (c) 4 (d) 2
36. 36.What is the remainder when 831 is divided by 12? (a) 2 (b) 4 (c) 3 (d) 9
37. 37.What is the remainder when 192 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 2
38. 38.What is the units digit of 3^17? (a) 9 (b) 1 (c) 3 (d) 7
39. 39.What is the smallest number you must add to 239 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
40. 40.Which of these numbers is divisible by 8? (a) 2487 (b) 2476 (c) 2490 (d) 2480

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