Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 192 · 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 units digit of 4^46? (a) 7 (b) 5 (c) 4 (d) 6
2. 2.What is the smallest number you must add to 144 to make it divisible by 11? (a) 9 (b) 8 (c) 1 (d) 10
3. 3.What is the smallest number you must add to 223 to make it divisible by 8? (a) 2 (b) 7 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 9? (a) 389 (b) 386 (c) 378 (d) 374
5. 5.What is the smallest number you must add to 611 to make it divisible by 10? (a) 1 (b) 9 (c) 7 (d) 8
6. 6.What is the remainder when 208 is divided by 11? (a) 1 (b) 10 (c) 9 (d) 8
7. 7.What is the remainder when 239 is divided by 3? (a) 0 (b) 1 (c) 2
8. 8.What is the remainder when 210 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
9. 9.Which of these numbers is divisible by 4? (a) 1778 (b) 1774 (c) 1769 (d) 1772
10. 10.What is the remainder when 201 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
11. 11.Which of these numbers is divisible by 6? (a) 2103 (b) 2091 (c) 2095 (d) 2094
12. 12.Which of these numbers is divisible by 3? (a) 2547 (b) 2545 (c) 2549 (d) 2552
13. 13.What is the smallest number you must add to 610 to make it divisible by 12? (a) 1 (b) 2 (c) 10 (d) 3
14. 14.Which of these numbers is divisible by 8? (a) 3605 (b) 3600 (c) 3593 (d) 3610
15. 15.Which of these numbers is divisible by 3? (a) 968 (b) 970 (c) 966 (d) 964
16. 16.Which of these numbers is divisible by 4? (a) 2109 (b) 2115 (c) 2108 (d) 2107
17. 17.What is the remainder when 171 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
18. 18.What is the units digit of 4^15? (a) 5 (b) 3 (c) 4 (d) 6
19. 19.Which of these numbers is divisible by 3? (a) 1893 (b) 1897 (c) 1891 (d) 1895
20. 20.What is the remainder when 258 is divided by 9? (a) 7 (b) 5 (c) 6 (d) 3
21. 21.Which of these numbers is divisible by 3? (a) 328 (b) 331 (c) 334 (d) 330
22. 22.What is the smallest number you must add to 854 to make it divisible by 11? (a) 4 (b) 3 (c) 5 (d) 7
23. 23.What is the units digit of 9^14? (a) 1 (b) 2 (c) 9 (d) 0
24. 24.What is the smallest number you must add to 684 to make it divisible by 10? (a) 7 (b) 6 (c) 5 (d) 4
25. 25.What is the smallest number you must add to 712 to make it divisible by 9? (a) 6 (b) 7 (c) 1 (d) 8
26. 26.What is the smallest number you must add to 598 to make it divisible by 10? (a) 3 (b) 1 (c) 2 (d) 8
27. 27.What is the remainder when 467 is divided by 6? (a) 3 (b) 1 (c) 5 (d) 4
28. 28.What is the units digit of 3^60? (a) 9 (b) 7 (c) 3 (d) 1
29. 29.What is the remainder when 857 is divided by 12? (a) 7 (b) 4 (c) 6 (d) 5
30. 30.What is the smallest number you must add to 166 to make it divisible by 4? (a) 2 (b) 3 (c) 1
31. 31.What is the remainder when 80 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 3
32. 32.What is the remainder when 111 is divided by 9? (a) 6 (b) 2 (c) 3 (d) 4
33. 33.What is the units digit of 9^48? (a) 1 (b) 2 (c) 9 (d) 0
34. 34.Which of these numbers is divisible by 9? (a) 6172 (b) 6189 (c) 6174 (d) 6177
35. 35.What is the remainder when 77 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 1
36. 36.What is the units digit of 3^10? (a) 9 (b) 3 (c) 7 (d) 1
37. 37.What is the remainder when 861 is divided by 11? (a) 3 (b) 4 (c) 8 (d) 2
38. 38.Which of these numbers is divisible by 9? (a) 7078 (b) 7065 (c) 7073 (d) 7061
39. 39.What is the smallest number you must add to 134 to make it divisible by 8? (a) 3 (b) 2 (c) 1 (d) 6
40. 40.Which of these numbers is divisible by 8? (a) 752 (b) 756 (c) 746 (d) 762

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