Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 705 · 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 units digit of 4^20? (a) 4 (b) 6 (c) 7 (d) 5
2. 2.Which of these numbers is divisible by 3? (a) 1840 (b) 1844 (c) 1837 (d) 1839
3. 3.Which of these numbers is divisible by 8? (a) 4929 (b) 4923 (c) 4920 (d) 4913
4. 4.Which of these numbers is divisible by 3? (a) 1285 (b) 1288 (c) 1282 (d) 1284
5. 5.What is the remainder when 142 is divided by 11? (a) 1 (b) 9 (c) 10 (d) 8
6. 6.What is the smallest number you must add to 478 to make it divisible by 7? (a) 4 (b) 2 (c) 6 (d) 5
7. 7.Which of these numbers is divisible by 11? (a) 7002 (b) 6994 (c) 6996 (d) 7012
8. 8.What is the remainder when 134 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
9. 9.What is the remainder when 673 is divided by 10? (a) 7 (b) 3 (c) 4 (d) 2
10. 10.Which of these numbers is divisible by 11? (a) 5871 (b) 5852 (c) 5855 (d) 5850
11. 11.What is the remainder when 127 is divided by 3? (a) 2 (b) 0 (c) 1
12. 12.What is the smallest number you must add to 475 to make it divisible by 9? (a) 3 (b) 1 (c) 2 (d) 7
13. 13.What is the units digit of 4^13? (a) 5 (b) 4 (c) 6 (d) 3
14. 14.Which of these numbers is divisible by 3? (a) 1186 (b) 1183 (c) 1189 (d) 1185
15. 15.Which of these numbers is divisible by 11? (a) 3854 (b) 3850 (c) 3843 (d) 3865
16. 16.What is the smallest number you must add to 183 to make it divisible by 4? (a) 3 (b) 2 (c) 1
17. 17.What is the units digit of 9^50? (a) 0 (b) 9 (c) 2 (d) 1
18. 18.What is the smallest number you must add to 369 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
19. 19.What is the smallest number you must add to 133 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
20. 20.What is the smallest number you must add to 169 to make it divisible by 9? (a) 3 (b) 7 (c) 1 (d) 2
21. 21.What is the remainder when 519 is divided by 9? (a) 6 (b) 7 (c) 3 (d) 5
22. 22.What is the units digit of 9^29? (a) 8 (b) 1 (c) 9 (d) 7
23. 23.Which of these numbers is divisible by 8? (a) 6596 (b) 6587 (c) 6605 (d) 6592
24. 24.What is the remainder when 235 is divided by 3? (a) 1 (b) 2 (c) 0
25. 25.What is the smallest number you must add to 137 to make it divisible by 3? (a) 2 (b) 1
26. 26.Which of these numbers is divisible by 9? (a) 2429 (b) 2433 (c) 2421 (d) 2420
27. 27.What is the remainder when 318 is divided by 11? (a) 1 (b) 10 (c) 8 (d) 9
28. 28.What is the units digit of 7^55? (a) 7 (b) 3 (c) 9 (d) 1
29. 29.What is the remainder when 328 is divided by 10? (a) 8 (b) 7 (c) 2 (d) 9
30. 30.What is the units digit of 9^10? (a) 0 (b) 2 (c) 1 (d) 9
31. 31.Which of these numbers is divisible by 9? (a) 5429 (b) 5422 (c) 5441 (d) 5427
32. 32.What is the units digit of 3^20? (a) 3 (b) 7 (c) 9 (d) 1
33. 33.Which of these numbers is divisible by 8? (a) 2411 (b) 2428 (c) 2416 (d) 2417
34. 34.What is the remainder when 165 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
35. 35.What is the units digit of 8^13? (a) 2 (b) 8 (c) 6 (d) 4
36. 36.What is the remainder when 333 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
37. 37.Which of these numbers is divisible by 4? (a) 910 (b) 906 (c) 904 (d) 902
38. 38.What is the smallest number you must add to 387 to make it divisible by 6? (a) 5 (b) 4 (c) 2 (d) 3
39. 39.Which of these numbers is divisible by 3? (a) 2379 (b) 2384 (c) 2381 (d) 2378
40. 40.What is the units digit of 8^23? (a) 6 (b) 2 (c) 8 (d) 4

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