Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 995 · 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 4? (a) 3566 (b) 3573 (c) 3570 (d) 3568
2. 2.What is the remainder when 520 is divided by 7? (a) 5 (b) 3 (c) 1 (d) 2
3. 3.What is the remainder when 352 is divided by 10? (a) 8 (b) 2 (c) 3 (d) 1
4. 4.What is the units digit of 2^44? (a) 6 (b) 8 (c) 4 (d) 2
5. 5.What is the smallest number you must add to 519 to make it divisible by 11? (a) 10 (b) 8 (c) 2 (d) 9
6. 6.What is the remainder when 172 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
7. 7.Which of these numbers is divisible by 9? (a) 901 (b) 909 (c) 920 (d) 912
8. 8.What is the remainder when 63 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 2
9. 9.What is the smallest number you must add to 637 to make it divisible by 12? (a) 1 (b) 9 (c) 10 (d) 11
10. 10.What is the smallest number you must add to 146 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the smallest number you must add to 157 to make it divisible by 8? (a) 5 (b) 2 (c) 4 (d) 3
12. 12.Which of these numbers is divisible by 4? (a) 2249 (b) 2253 (c) 2258 (d) 2252
13. 13.What is the smallest number you must add to 53 to make it divisible by 3? (a) 2 (b) 1
14. 14.What is the units digit of 2^53? (a) 4 (b) 6 (c) 2 (d) 8
15. 15.Which of these numbers is divisible by 11? (a) 880 (b) 879 (c) 895 (d) 890
16. 16.Which of these numbers is divisible by 9? (a) 522 (b) 528 (c) 538 (d) 519
17. 17.What is the remainder when 387 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
18. 18.What is the remainder when 68 is divided by 3? (a) 2 (b) 1 (c) 0
19. 19.What is the remainder when 325 is divided by 7? (a) 3 (b) 5 (c) 2 (d) 4
20. 20.What is the remainder when 526 is divided by 11? (a) 8 (b) 2 (c) 10 (d) 9
21. 21.What is the smallest number you must add to 146 to make it divisible by 12? (a) 2 (b) 10 (c) 9 (d) 11
22. 22.What is the units digit of 7^57? (a) 9 (b) 3 (c) 7 (d) 1
23. 23.What is the units digit of 9^19? (a) 7 (b) 1 (c) 8 (d) 9
24. 24.What is the remainder when 520 is divided by 12? (a) 3 (b) 4 (c) 5 (d) 8
25. 25.What is the units digit of 3^15? (a) 9 (b) 1 (c) 7 (d) 3
26. 26.What is the smallest number you must add to 175 to make it divisible by 3? (a) 2 (b) 1
27. 27.What is the smallest number you must add to 172 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
28. 28.Which of these numbers is divisible by 8? (a) 5506 (b) 5496 (c) 5499 (d) 5490
29. 29.What is the smallest number you must add to 344 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
30. 30.What is the remainder when 166 is divided by 11? (a) 2 (b) 10 (c) 0 (d) 1
31. 31.What is the units digit of 7^46? (a) 1 (b) 9 (c) 7 (d) 3
32. 32.What is the units digit of 9^15? (a) 9 (b) 7 (c) 1 (d) 8
33. 33.Which of these numbers is divisible by 6? (a) 1628 (b) 1620 (c) 1625 (d) 1619
34. 34.What is the units digit of 7^60? (a) 3 (b) 7 (c) 1 (d) 9
35. 35.Which of these numbers is divisible by 3? (a) 1732 (b) 1736 (c) 1738 (d) 1734
36. 36.What is the smallest number you must add to 252 to make it divisible by 8? (a) 6 (b) 4 (c) 5 (d) 3
37. 37.What is the remainder when 325 is divided by 8? (a) 6 (b) 3 (c) 5 (d) 4
38. 38.What is the remainder when 460 is divided by 9? (a) 8 (b) 2 (c) 1 (d) 0
39. 39.Which of these numbers is divisible by 9? (a) 1989 (b) 1990 (c) 2001 (d) 1985
40. 40.What is the smallest number you must add to 123 to make it divisible by 11? (a) 2 (b) 8 (c) 9 (d) 10

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