Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 133 · 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 111 to make it divisible by 11? (a) 8 (b) 9 (c) 1 (d) 10
2. 2.What is the remainder when 301 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
3. 3.What is the units digit of 9^34? (a) 0 (b) 9 (c) 2 (d) 1
4. 4.What is the remainder when 142 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
5. 5.Which of these numbers is divisible by 4? (a) 1113 (b) 1108 (c) 1107 (d) 1109
6. 6.What is the units digit of 7^17? (a) 9 (b) 3 (c) 1 (d) 7
7. 7.What is the units digit of 4^21? (a) 3 (b) 4 (c) 6 (d) 5
8. 8.What is the smallest number you must add to 66 to make it divisible by 4? (a) 2 (b) 1 (c) 3
9. 9.Which of these numbers is divisible by 11? (a) 4775 (b) 4765 (c) 4793 (d) 4774
10. 10.What is the smallest number you must add to 86 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
11. 11.What is the units digit of 4^17? (a) 6 (b) 4 (c) 5 (d) 3
12. 12.Which of these numbers is divisible by 8? (a) 4976 (b) 4980 (c) 4990 (d) 4975
13. 13.What is the remainder when 242 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
14. 14.What is the remainder when 353 is divided by 10? (a) 4 (b) 3 (c) 2 (d) 7
15. 15.What is the remainder when 146 is divided by 12? (a) 10 (b) 3 (c) 1 (d) 2
16. 16.Which of these numbers is divisible by 4? (a) 3157 (b) 3165 (c) 3160 (d) 3161
17. 17.What is the remainder when 275 is divided by 7? (a) 3 (b) 2 (c) 1 (d) 5
18. 18.Which of these numbers is divisible by 11? (a) 6535 (b) 6545 (c) 6549 (d) 6557
19. 19.What is the units digit of 8^17? (a) 4 (b) 6 (c) 2 (d) 8
20. 20.What is the smallest number you must add to 354 to make it divisible by 7? (a) 3 (b) 4 (c) 2 (d) 5
21. 21.What is the units digit of 4^22? (a) 7 (b) 4 (c) 6 (d) 5
22. 22.What is the remainder when 61 is divided by 3? (a) 0 (b) 1 (c) 2
23. 23.Which of these numbers is divisible by 3? (a) 604 (b) 602 (c) 607 (d) 603
24. 24.What is the smallest number you must add to 203 to make it divisible by 3? (a) 1 (b) 2
25. 25.Which of these numbers is divisible by 9? (a) 3074 (b) 3082 (c) 3061 (d) 3069
26. 26.What is the smallest number you must add to 320 to make it divisible by 9? (a) 6 (b) 4 (c) 3 (d) 5
27. 27.What is the units digit of 2^31? (a) 6 (b) 2 (c) 4 (d) 8
28. 28.What is the smallest number you must add to 146 to make it divisible by 3? (a) 1 (b) 2
29. 29.What is the remainder when 174 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 1
30. 30.What is the smallest number you must add to 482 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
31. 31.What is the smallest number you must add to 222 to make it divisible by 9? (a) 3 (b) 4 (c) 2 (d) 6
32. 32.What is the remainder when 177 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
33. 33.What is the smallest number you must add to 263 to make it divisible by 11? (a) 3 (b) 1 (c) 2 (d) 10
34. 34.Which of these numbers is divisible by 6? (a) 4380 (b) 4384 (c) 4388 (d) 4379
35. 35.What is the units digit of 4^32? (a) 6 (b) 7 (c) 5 (d) 4
36. 36.What is the units digit of 8^50? (a) 6 (b) 4 (c) 8 (d) 2
37. 37.What is the units digit of 4^45? (a) 5 (b) 4 (c) 6 (d) 3
38. 38.What is the remainder when 131 is divided by 9? (a) 7 (b) 6 (c) 5 (d) 4
39. 39.What is the units digit of 2^51? (a) 8 (b) 6 (c) 4 (d) 2
40. 40.What is the units digit of 3^43? (a) 1 (b) 7 (c) 3 (d) 9

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