Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 294 · 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 smallest number you must add to 61 to make it divisible by 4? (a) 2 (b) 3 (c) 1
2. 2.Which of these numbers is divisible by 11? (a) 9211 (b) 9191 (c) 9197 (d) 9196
3. 3.What is the units digit of 7^21? (a) 9 (b) 1 (c) 3 (d) 7
4. 4.What is the units digit of 3^11? (a) 1 (b) 9 (c) 7 (d) 3
5. 5.What is the smallest number you must add to 106 to make it divisible by 4? (a) 1 (b) 3 (c) 2
6. 6.What is the remainder when 134 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
7. 7.What is the smallest number you must add to 213 to make it divisible by 4? (a) 2 (b) 3 (c) 1
8. 8.What is the smallest number you must add to 226 to make it divisible by 9? (a) 8 (b) 1 (c) 6 (d) 7
9. 9.What is the units digit of 2^29? (a) 4 (b) 6 (c) 8 (d) 2
10. 10.What is the remainder when 106 is divided by 9? (a) 2 (b) 6 (c) 8 (d) 7
11. 11.What is the units digit of 7^23? (a) 1 (b) 3 (c) 9 (d) 7
12. 12.What is the remainder when 216 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
13. 13.Which of these numbers is divisible by 6? (a) 4692 (b) 4702 (c) 4690 (d) 4693
14. 14.What is the smallest number you must add to 443 to make it divisible by 6? (a) 1 (b) 3 (c) 5 (d) 2
15. 15.Which of these numbers is divisible by 9? (a) 3194 (b) 3177 (c) 3171 (d) 3179
16. 16.Which of these numbers is divisible by 8? (a) 696 (b) 693 (c) 709 (d) 697
17. 17.What is the remainder when 49 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
18. 18.Which of these numbers is divisible by 6? (a) 274 (b) 266 (c) 270 (d) 277
19. 19.What is the units digit of 8^22? (a) 8 (b) 4 (c) 2 (d) 6
20. 20.What is the units digit of 4^12? (a) 5 (b) 4 (c) 7 (d) 6
21. 21.What is the remainder when 125 is divided by 3? (a) 0 (b) 2 (c) 1
22. 22.What is the units digit of 8^56? (a) 2 (b) 8 (c) 6 (d) 4
23. 23.What is the smallest number you must add to 51 to make it divisible by 4? (a) 2 (b) 3 (c) 1
24. 24.What is the units digit of 9^56? (a) 9 (b) 2 (c) 0 (d) 1
25. 25.What is the remainder when 557 is divided by 9? (a) 1 (b) 8 (c) 6 (d) 7
26. 26.Which of these numbers is divisible by 11? (a) 9817 (b) 9831 (c) 9839 (d) 9823
27. 27.What is the remainder when 356 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 0
28. 28.Which of these numbers is divisible by 4? (a) 279 (b) 273 (c) 283 (d) 276
29. 29.Which of these numbers is divisible by 8? (a) 5513 (b) 5512 (c) 5525 (d) 5511
30. 30.What is the smallest number you must add to 434 to make it divisible by 11? (a) 6 (b) 8 (c) 7 (d) 5
31. 31.What is the remainder when 89 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
32. 32.What is the smallest number you must add to 469 to make it divisible by 9? (a) 8 (b) 6 (c) 7 (d) 1
33. 33.What is the units digit of 3^10? (a) 1 (b) 7 (c) 3 (d) 9
34. 34.What is the remainder when 705 is divided by 10? (a) 6 (b) 7 (c) 5 (d) 4
35. 35.Which of these numbers is divisible by 8? (a) 4184 (b) 4180 (c) 4191 (d) 4195
36. 36.Which of these numbers is divisible by 9? (a) 7580 (b) 7601 (c) 7587 (d) 7590
37. 37.What is the units digit of 7^24? (a) 7 (b) 9 (c) 3 (d) 1
38. 38.Which of these numbers is divisible by 6? (a) 5401 (b) 5390 (c) 5395 (d) 5394
39. 39.What is the remainder when 131 is divided by 10? (a) 2 (b) 1 (c) 0 (d) 9
40. 40.What is the remainder when 793 is divided by 11? (a) 0 (b) 2 (c) 1 (d) 10

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