Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 913 · 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 233 is divided by 7? (a) 5 (b) 3 (c) 2 (d) 1
2. 2.Which of these numbers is divisible by 9? (a) 1185 (b) 1179 (c) 1178 (d) 1190
3. 3.What is the smallest number you must add to 708 to make it divisible by 10? (a) 2 (b) 8 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 11? (a) 1396 (b) 1418 (c) 1398 (d) 1397
5. 5.What is the smallest number you must add to 346 to make it divisible by 9? (a) 4 (b) 6 (c) 7 (d) 5
6. 6.What is the units digit of 3^21? (a) 9 (b) 7 (c) 1 (d) 3
7. 7.Which of these numbers is divisible by 11? (a) 8778 (b) 8771 (c) 8794 (d) 8786
8. 8.Which of these numbers is divisible by 11? (a) 6393 (b) 6402 (c) 6404 (d) 6416
9. 9.Which of these numbers is divisible by 11? (a) 9453 (b) 9463 (c) 9476 (d) 9460
10. 10.What is the units digit of 8^23? (a) 2 (b) 4 (c) 6 (d) 8
11. 11.What is the units digit of 9^14? (a) 9 (b) 2 (c) 0 (d) 1
12. 12.What is the units digit of 7^58? (a) 7 (b) 3 (c) 1 (d) 9
13. 13.Which of these numbers is divisible by 6? (a) 3176 (b) 3170 (c) 3174 (d) 3184
14. 14.Which of these numbers is divisible by 4? (a) 1319 (b) 1322 (c) 1320 (d) 1325
15. 15.Which of these numbers is divisible by 6? (a) 1620 (b) 1621 (c) 1629 (d) 1617
16. 16.What is the units digit of 8^25? (a) 6 (b) 8 (c) 4 (d) 2
17. 17.Which of these numbers is divisible by 8? (a) 5059 (b) 5054 (c) 5048 (d) 5046
18. 18.What is the units digit of 8^45? (a) 2 (b) 8 (c) 4 (d) 6
19. 19.What is the remainder when 363 is divided by 12? (a) 2 (b) 9 (c) 4 (d) 3
20. 20.What is the remainder when 218 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
21. 21.What is the smallest number you must add to 154 to make it divisible by 10? (a) 4 (b) 7 (c) 6 (d) 5
22. 22.What is the units digit of 2^50? (a) 6 (b) 2 (c) 4 (d) 8
23. 23.What is the smallest number you must add to 617 to make it divisible by 10? (a) 2 (b) 3 (c) 7 (d) 4
24. 24.Which of these numbers is divisible by 3? (a) 2071 (b) 2069 (c) 2067 (d) 2065
25. 25.Which of these numbers is divisible by 9? (a) 2461 (b) 2454 (c) 2448 (d) 2442
26. 26.What is the remainder when 104 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
27. 27.What is the smallest number you must add to 318 to make it divisible by 10? (a) 8 (b) 3 (c) 1 (d) 2
28. 28.What is the remainder when 228 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
29. 29.What is the units digit of 7^23? (a) 9 (b) 7 (c) 3 (d) 1
30. 30.Which of these numbers is divisible by 11? (a) 5168 (b) 5172 (c) 5185 (d) 5170
31. 31.What is the units digit of 3^56? (a) 1 (b) 3 (c) 9 (d) 7
32. 32.What is the smallest number you must add to 87 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
33. 33.What is the units digit of 2^18? (a) 2 (b) 8 (c) 6 (d) 4
34. 34.Which of these numbers is divisible by 8? (a) 1877 (b) 1884 (c) 1872 (d) 1868
35. 35.Which of these numbers is divisible by 9? (a) 5949 (b) 5943 (c) 5966 (d) 5956
36. 36.Which of these numbers is divisible by 4? (a) 1575 (b) 1582 (c) 1576 (d) 1577
37. 37.What is the units digit of 8^44? (a) 4 (b) 8 (c) 2 (d) 6
38. 38.What is the remainder when 575 is divided by 8? (a) 1 (b) 6 (c) 5 (d) 7
39. 39.What is the remainder when 40 is divided by 3? (a) 0 (b) 2 (c) 1
40. 40.What is the remainder when 156 is divided by 8? (a) 6 (b) 4 (c) 5 (d) 3

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