Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 488 · 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 468 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6
2. 2.Which of these numbers is divisible by 6? (a) 1842 (b) 1840 (c) 1843 (d) 1852
3. 3.Which of these numbers is divisible by 6? (a) 1147 (b) 1155 (c) 1161 (d) 1152
4. 4.What is the smallest number you must add to 111 to make it divisible by 10? (a) 1 (b) 9 (c) 7 (d) 8
5. 5.Which of these numbers is divisible by 9? (a) 2286 (b) 2299 (c) 2287 (d) 2279
6. 6.What is the remainder when 353 is divided by 10? (a) 4 (b) 2 (c) 3 (d) 7
7. 7.Which of these numbers is divisible by 3? (a) 2126 (b) 2129 (c) 2132 (d) 2127
8. 8.What is the smallest number you must add to 345 to make it divisible by 8? (a) 6 (b) 7 (c) 1 (d) 5
9. 9.What is the smallest number you must add to 106 to make it divisible by 3? (a) 1 (b) 2
10. 10.What is the units digit of 7^16? (a) 9 (b) 1 (c) 3 (d) 7
11. 11.What is the smallest number you must add to 322 to make it divisible by 9? (a) 3 (b) 1 (c) 7 (d) 2
12. 12.What is the remainder when 100 is divided by 7? (a) 2 (b) 5 (c) 3 (d) 1
13. 13.Which of these numbers is divisible by 3? (a) 273 (b) 278 (c) 274 (d) 271
14. 14.What is the smallest number you must add to 546 to make it divisible by 10? (a) 3 (b) 6 (c) 5 (d) 4
15. 15.What is the smallest number you must add to 181 to make it divisible by 4? (a) 2 (b) 1 (c) 3
16. 16.What is the remainder when 335 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6
17. 17.What is the units digit of 4^52? (a) 4 (b) 5 (c) 7 (d) 6
18. 18.What is the units digit of 9^21? (a) 1 (b) 7 (c) 9 (d) 8
19. 19.What is the remainder when 490 is divided by 8? (a) 3 (b) 6 (c) 1 (d) 2
20. 20.What is the units digit of 7^28? (a) 3 (b) 7 (c) 9 (d) 1
21. 21.What is the smallest number you must add to 643 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 6
22. 22.What is the units digit of 9^29? (a) 7 (b) 9 (c) 8 (d) 1
23. 23.Which of these numbers is divisible by 4? (a) 713 (b) 706 (c) 708 (d) 711
24. 24.What is the remainder when 215 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
25. 25.What is the remainder when 369 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
26. 26.Which of these numbers is divisible by 4? (a) 1096 (b) 1095 (c) 1098 (d) 1101
27. 27.What is the smallest number you must add to 119 to make it divisible by 6? (a) 2 (b) 3 (c) 1 (d) 5
28. 28.What is the smallest number you must add to 607 to make it divisible by 8? (a) 3 (b) 2 (c) 7 (d) 1
29. 29.What is the units digit of 3^12? (a) 9 (b) 1 (c) 3 (d) 7
30. 30.What is the units digit of 7^33? (a) 1 (b) 9 (c) 3 (d) 7
31. 31.What is the remainder when 68 is divided by 3? (a) 0 (b) 1 (c) 2
32. 32.Which of these numbers is divisible by 6? (a) 3483 (b) 3489 (c) 3475 (d) 3480
33. 33.What is the units digit of 2^12? (a) 2 (b) 6 (c) 4 (d) 8
34. 34.What is the units digit of 9^17? (a) 8 (b) 1 (c) 9 (d) 7
35. 35.What is the remainder when 49 is divided by 3? (a) 0 (b) 2 (c) 1
36. 36.What is the remainder when 382 is divided by 11? (a) 3 (b) 9 (c) 8 (d) 7
37. 37.What is the units digit of 8^59? (a) 2 (b) 4 (c) 8 (d) 6
38. 38.Which of these numbers is divisible by 3? (a) 194 (b) 189 (c) 191 (d) 187
39. 39.What is the smallest number you must add to 247 to make it divisible by 11? (a) 8 (b) 5 (c) 6 (d) 7
40. 40.What is the units digit of 8^42? (a) 6 (b) 2 (c) 4 (d) 8

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