Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 930 · 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 remainder when 565 is divided by 9? (a) 2 (b) 6 (c) 7 (d) 8
2. 2.What is the units digit of 3^34? (a) 7 (b) 1 (c) 9 (d) 3
3. 3.Which of these numbers is divisible by 9? (a) 3965 (b) 3955 (c) 3950 (d) 3951
4. 4.What is the remainder when 561 is divided by 7? (a) 6 (b) 0 (c) 1 (d) 2
5. 5.Which of these numbers is divisible by 11? (a) 9447 (b) 9436 (c) 9453 (d) 9438
6. 6.What is the smallest number you must add to 226 to make it divisible by 3? (a) 2 (b) 1
7. 7.What is the smallest number you must add to 196 to make it divisible by 10? (a) 3 (b) 4 (c) 5 (d) 6
8. 8.What is the smallest number you must add to 148 to make it divisible by 12? (a) 7 (b) 4 (c) 9 (d) 8
9. 9.What is the units digit of 7^34? (a) 7 (b) 1 (c) 9 (d) 3
10. 10.What is the remainder when 394 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
11. 11.What is the smallest number you must add to 920 to make it divisible by 12? (a) 3 (b) 5 (c) 4 (d) 8
12. 12.Which of these numbers is divisible by 6? (a) 602 (b) 607 (c) 606 (d) 614
13. 13.What is the smallest number you must add to 88 to make it divisible by 7? (a) 5 (b) 2 (c) 3 (d) 4
14. 14.What is the remainder when 42 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
15. 15.Which of these numbers is divisible by 8? (a) 5210 (b) 5208 (c) 5207 (d) 5220
16. 16.Which of these numbers is divisible by 4? (a) 2218 (b) 2220 (c) 2225 (d) 2223
17. 17.What is the remainder when 197 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
18. 18.What is the remainder when 196 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
19. 19.Which of these numbers is divisible by 3? (a) 208 (b) 211 (c) 215 (d) 210
20. 20.What is the remainder when 386 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2
21. 21.What is the units digit of 2^59? (a) 6 (b) 8 (c) 4 (d) 2
22. 22.What is the units digit of 7^12? (a) 9 (b) 3 (c) 7 (d) 1
23. 23.What is the units digit of 3^28? (a) 1 (b) 3 (c) 9 (d) 7
24. 24.What is the smallest number you must add to 131 to make it divisible by 4? (a) 3 (b) 2 (c) 1
25. 25.Which of these numbers is divisible by 3? (a) 502 (b) 500 (c) 498 (d) 497
26. 26.What is the units digit of 8^23? (a) 4 (b) 6 (c) 8 (d) 2
27. 27.What is the remainder when 143 is divided by 3? (a) 1 (b) 0 (c) 2
28. 28.What is the remainder when 706 is divided by 10? (a) 4 (b) 7 (c) 5 (d) 6
29. 29.What is the smallest number you must add to 213 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
30. 30.What is the units digit of 4^34? (a) 7 (b) 4 (c) 6 (d) 5
31. 31.Which of these numbers is divisible by 9? (a) 516 (b) 522 (c) 528 (d) 539
32. 32.What is the remainder when 366 is divided by 8? (a) 2 (b) 6 (c) 7 (d) 5
33. 33.What is the units digit of 9^21? (a) 9 (b) 7 (c) 8 (d) 1
34. 34.Which of these numbers is divisible by 3? (a) 554 (b) 556 (c) 560 (d) 555
35. 35.What is the smallest number you must add to 124 to make it divisible by 12? (a) 8 (b) 4 (c) 9 (d) 7
36. 36.What is the smallest number you must add to 61 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
37. 37.What is the units digit of 8^58? (a) 6 (b) 2 (c) 8 (d) 4
38. 38.What is the smallest number you must add to 143 to make it divisible by 8? (a) 7 (b) 1 (c) 3 (d) 2
39. 39.What is the smallest number you must add to 108 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
40. 40.What is the smallest number you must add to 301 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1

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