Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 752 · 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 2^26? (a) 8 (b) 6 (c) 2 (d) 4
2. 2.What is the smallest number you must add to 221 to make it divisible by 9? (a) 4 (b) 5 (c) 6 (d) 3
3. 3.Which of these numbers is divisible by 3? (a) 1135 (b) 1131 (c) 1132 (d) 1129
4. 4.Which of these numbers is divisible by 9? (a) 4462 (b) 4472 (c) 4450 (d) 4455
5. 5.Which of these numbers is divisible by 6? (a) 2112 (b) 2110 (c) 2119 (d) 2116
6. 6.What is the remainder when 215 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
7. 7.What is the units digit of 8^31? (a) 8 (b) 2 (c) 4 (d) 6
8. 8.Which of these numbers is divisible by 3? (a) 2626 (b) 2620 (c) 2623 (d) 2622
9. 9.What is the smallest number you must add to 611 to make it divisible by 11? (a) 5 (b) 6 (c) 4 (d) 7
10. 10.What is the smallest number you must add to 134 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
11. 11.What is the remainder when 215 is divided by 3? (a) 2 (b) 1 (c) 0
12. 12.What is the smallest number you must add to 140 to make it divisible by 3? (a) 1 (b) 2
13. 13.What is the units digit of 2^50? (a) 4 (b) 8 (c) 2 (d) 6
14. 14.What is the smallest number you must add to 58 to make it divisible by 4? (a) 3 (b) 1 (c) 2
15. 15.What is the remainder when 215 is divided by 12? (a) 1 (b) 11 (c) 10 (d) 9
16. 16.What is the smallest number you must add to 426 to make it divisible by 8? (a) 2 (b) 6 (c) 7 (d) 5
17. 17.What is the smallest number you must add to 421 to make it divisible by 7? (a) 1 (b) 4 (c) 5 (d) 6
18. 18.What is the smallest number you must add to 170 to make it divisible by 3? (a) 1 (b) 2
19. 19.What is the remainder when 148 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
20. 20.What is the smallest number you must add to 296 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
21. 21.What is the units digit of 9^48? (a) 9 (b) 0 (c) 1 (d) 2
22. 22.What is the smallest number you must add to 449 to make it divisible by 8? (a) 5 (b) 7 (c) 6 (d) 1
23. 23.Which of these numbers is divisible by 8? (a) 5535 (b) 5538 (c) 5536 (d) 5548
24. 24.What is the units digit of 4^16? (a) 4 (b) 5 (c) 6 (d) 7
25. 25.Which of these numbers is divisible by 6? (a) 3555 (b) 3562 (c) 3568 (d) 3558
26. 26.What is the remainder when 94 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
27. 27.What is the units digit of 9^56? (a) 9 (b) 2 (c) 1 (d) 0
28. 28.What is the units digit of 9^15? (a) 1 (b) 7 (c) 8 (d) 9
29. 29.What is the remainder when 61 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
30. 30.Which of these numbers is divisible by 4? (a) 387 (b) 384 (c) 391 (d) 383
31. 31.Which of these numbers is divisible by 4? (a) 3385 (b) 3384 (c) 3383 (d) 3391
32. 32.What is the smallest number you must add to 56 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
33. 33.What is the units digit of 8^55? (a) 6 (b) 8 (c) 2 (d) 4
34. 34.What is the remainder when 170 is divided by 11? (a) 5 (b) 4 (c) 6 (d) 7
35. 35.Which of these numbers is divisible by 4? (a) 2115 (b) 2112 (c) 2117 (d) 2109
36. 36.What is the units digit of 2^33? (a) 8 (b) 4 (c) 6 (d) 2
37. 37.Which of these numbers is divisible by 11? (a) 9867 (b) 9864 (c) 9868 (d) 9880
38. 38.Which of these numbers is divisible by 11? (a) 5572 (b) 5561 (c) 5545 (d) 5555
39. 39.What is the units digit of 7^59? (a) 7 (b) 1 (c) 3 (d) 9
40. 40.Which of these numbers is divisible by 8? (a) 6915 (b) 6901 (c) 6904 (d) 6911

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