Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 209 · 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 9? (a) 6115 (b) 6111 (c) 6106 (d) 6125
2. 2.What is the remainder when 453 is divided by 7? (a) 6 (b) 5 (c) 2 (d) 4
3. 3.What is the remainder when 241 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 2
4. 4.What is the smallest number you must add to 194 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
5. 5.What is the smallest number you must add to 236 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3
6. 6.Which of these numbers is divisible by 11? (a) 2988 (b) 2970 (c) 2964 (d) 2972
7. 7.What is the smallest number you must add to 441 to make it divisible by 10? (a) 7 (b) 1 (c) 9 (d) 8
8. 8.What is the units digit of 9^17? (a) 9 (b) 7 (c) 8 (d) 1
9. 9.What is the remainder when 625 is divided by 12? (a) 11 (b) 0 (c) 2 (d) 1
10. 10.What is the units digit of 4^45? (a) 6 (b) 5 (c) 4 (d) 3
11. 11.What is the smallest number you must add to 321 to make it divisible by 12? (a) 4 (b) 9 (c) 3 (d) 2
12. 12.Which of these numbers is divisible by 11? (a) 5129 (b) 5126 (c) 5141 (d) 5123
13. 13.What is the units digit of 3^35? (a) 9 (b) 1 (c) 7 (d) 3
14. 14.What is the remainder when 59 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
15. 15.What is the smallest number you must add to 136 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
16. 16.What is the smallest number you must add to 139 to make it divisible by 4? (a) 3 (b) 1 (c) 2
17. 17.What is the units digit of 8^37? (a) 6 (b) 2 (c) 8 (d) 4
18. 18.Which of these numbers is divisible by 4? (a) 512 (b) 518 (c) 509 (d) 513
19. 19.What is the remainder when 98 is divided by 8? (a) 3 (b) 2 (c) 1 (d) 6
20. 20.What is the units digit of 2^11? (a) 8 (b) 6 (c) 4 (d) 2
21. 21.What is the remainder when 316 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
22. 22.What is the smallest number you must add to 105 to make it divisible by 4? (a) 3 (b) 1 (c) 2
23. 23.What is the units digit of 2^30? (a) 2 (b) 8 (c) 4 (d) 6
24. 24.Which of these numbers is divisible by 8? (a) 2681 (b) 2675 (c) 2693 (d) 2680
25. 25.Which of these numbers is divisible by 4? (a) 165 (b) 169 (c) 173 (d) 168
26. 26.Which of these numbers is divisible by 9? (a) 1876 (b) 1864 (c) 1859 (d) 1863
27. 27.What is the units digit of 3^43? (a) 9 (b) 1 (c) 7 (d) 3
28. 28.What is the smallest number you must add to 298 to make it divisible by 6? (a) 2 (b) 4 (c) 1 (d) 3
29. 29.What is the remainder when 754 is divided by 11? (a) 7 (b) 8 (c) 5 (d) 6
30. 30.What is the remainder when 311 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
31. 31.What is the units digit of 3^49? (a) 7 (b) 3 (c) 1 (d) 9
32. 32.What is the units digit of 3^48? (a) 3 (b) 7 (c) 9 (d) 1
33. 33.Which of these numbers is divisible by 3? (a) 164 (b) 160 (c) 158 (d) 159
34. 34.What is the units digit of 8^38? (a) 2 (b) 6 (c) 8 (d) 4
35. 35.What is the units digit of 2^38? (a) 2 (b) 8 (c) 4 (d) 6
36. 36.What is the units digit of 9^45? (a) 8 (b) 7 (c) 9 (d) 1
37. 37.Which of these numbers is divisible by 8? (a) 5259 (b) 5271 (c) 5256 (d) 5255
38. 38.Which of these numbers is divisible by 11? (a) 8436 (b) 8440 (c) 8437 (d) 8450
39. 39.What is the units digit of 9^22? (a) 0 (b) 2 (c) 1 (d) 9
40. 40.What is the remainder when 109 is divided by 8? (a) 4 (b) 6 (c) 5 (d) 3

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