Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 536 · 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 258 is divided by 7? (a) 4 (b) 6 (c) 1 (d) 5
2. 2.What is the smallest number you must add to 128 to make it divisible by 3? (a) 2 (b) 1
3. 3.What is the units digit of 7^31? (a) 7 (b) 1 (c) 3 (d) 9
4. 4.What is the remainder when 161 is divided by 8? (a) 1 (b) 0 (c) 7 (d) 2
5. 5.Which of these numbers is divisible by 9? (a) 3474 (b) 3486 (c) 3481 (d) 3469
6. 6.What is the smallest number you must add to 335 to make it divisible by 8? (a) 1 (b) 7 (c) 3 (d) 2
7. 7.What is the units digit of 2^52? (a) 6 (b) 8 (c) 2 (d) 4
8. 8.What is the remainder when 536 is divided by 12? (a) 8 (b) 7 (c) 4 (d) 9
9. 9.Which of these numbers is divisible by 8? (a) 4200 (b) 4210 (c) 4205 (d) 4196
10. 10.What is the remainder when 195 is divided by 8? (a) 5 (b) 2 (c) 3 (d) 4
11. 11.Which of these numbers is divisible by 3? (a) 1213 (b) 1217 (c) 1219 (d) 1215
12. 12.What is the remainder when 770 is divided by 12? (a) 3 (b) 2 (c) 1 (d) 10
13. 13.What is the remainder when 360 is divided by 7? (a) 2 (b) 5 (c) 3 (d) 4
14. 14.What is the units digit of 2^20? (a) 4 (b) 8 (c) 2 (d) 6
15. 15.Which of these numbers is divisible by 3? (a) 316 (b) 313 (c) 315 (d) 319
16. 16.Which of these numbers is divisible by 8? (a) 4070 (b) 4064 (c) 4060 (d) 4079
17. 17.Which of these numbers is divisible by 3? (a) 1109 (b) 1111 (c) 1114 (d) 1110
18. 18.Which of these numbers is divisible by 11? (a) 1745 (b) 1751 (c) 1738 (d) 1734
19. 19.What is the units digit of 2^40? (a) 4 (b) 2 (c) 6 (d) 8
20. 20.What is the smallest number you must add to 177 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 1
21. 21.What is the remainder when 126 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
22. 22.What is the units digit of 3^34? (a) 3 (b) 7 (c) 1 (d) 9
23. 23.What is the smallest number you must add to 224 to make it divisible by 10? (a) 7 (b) 5 (c) 6 (d) 4
24. 24.What is the smallest number you must add to 140 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 5
25. 25.What is the smallest number you must add to 274 to make it divisible by 11? (a) 1 (b) 2 (c) 10 (d) 3
26. 26.What is the remainder when 532 is divided by 11? (a) 5 (b) 7 (c) 4 (d) 3
27. 27.Which of these numbers is divisible by 8? (a) 1759 (b) 1763 (c) 1760 (d) 1773
28. 28.What is the remainder when 282 is divided by 8? (a) 2 (b) 1 (c) 6 (d) 3
29. 29.Which of these numbers is divisible by 4? (a) 389 (b) 381 (c) 384 (d) 385
30. 30.What is the units digit of 3^29? (a) 9 (b) 1 (c) 3 (d) 7
31. 31.Which of these numbers is divisible by 6? (a) 3782 (b) 3786 (c) 3793 (d) 3789
32. 32.What is the smallest number you must add to 687 to make it divisible by 12? (a) 9 (b) 10 (c) 3 (d) 8
33. 33.What is the smallest number you must add to 274 to make it divisible by 6? (a) 1 (b) 4 (c) 2 (d) 3
34. 34.What is the smallest number you must add to 665 to make it divisible by 11? (a) 6 (b) 5 (c) 7 (d) 8
35. 35.What is the units digit of 3^16? (a) 3 (b) 1 (c) 9 (d) 7
36. 36.Which of these numbers is divisible by 9? (a) 1873 (b) 1884 (c) 1866 (d) 1872
37. 37.What is the units digit of 2^21? (a) 6 (b) 4 (c) 8 (d) 2
38. 38.What is the smallest number you must add to 124 to make it divisible by 12? (a) 7 (b) 4 (c) 9 (d) 8
39. 39.What is the remainder when 55 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
40. 40.What is the remainder when 88 is divided by 3? (a) 2 (b) 0 (c) 1

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