Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 681 · 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 3^51? (a) 9 (b) 7 (c) 1 (d) 3
2. 2.What is the smallest number you must add to 103 to make it divisible by 7? (a) 2 (b) 3 (c) 5 (d) 1
3. 3.What is the units digit of 3^14? (a) 7 (b) 3 (c) 1 (d) 9
4. 4.What is the remainder when 927 is divided by 12? (a) 4 (b) 2 (c) 9 (d) 3
5. 5.What is the remainder when 155 is divided by 11? (a) 10 (b) 2 (c) 0 (d) 1
6. 6.What is the smallest number you must add to 514 to make it divisible by 9? (a) 6 (b) 7 (c) 1 (d) 8
7. 7.What is the units digit of 8^26? (a) 8 (b) 6 (c) 2 (d) 4
8. 8.What is the remainder when 200 is divided by 11? (a) 9 (b) 3 (c) 1 (d) 2
9. 9.What is the remainder when 454 is divided by 11? (a) 8 (b) 2 (c) 4 (d) 3
10. 10.What is the smallest number you must add to 267 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
11. 11.Which of these numbers is divisible by 3? (a) 2383 (b) 2387 (c) 2380 (d) 2382
12. 12.What is the units digit of 3^34? (a) 1 (b) 3 (c) 7 (d) 9
13. 13.Which of these numbers is divisible by 8? (a) 5878 (b) 5861 (c) 5864 (d) 5867
14. 14.Which of these numbers is divisible by 4? (a) 250 (b) 255 (c) 252 (d) 257
15. 15.What is the smallest number you must add to 476 to make it divisible by 10? (a) 4 (b) 6 (c) 5 (d) 3
16. 16.What is the remainder when 404 is divided by 6? (a) 2 (b) 4 (c) 1 (d) 3
17. 17.What is the remainder when 311 is divided by 8? (a) 1 (b) 5 (c) 7 (d) 6
18. 18.What is the units digit of 2^58? (a) 8 (b) 4 (c) 6 (d) 2
19. 19.What is the smallest number you must add to 234 to make it divisible by 12? (a) 7 (b) 6 (c) 5 (d) 8
20. 20.What is the smallest number you must add to 562 to make it divisible by 8? (a) 5 (b) 6 (c) 7 (d) 2
21. 21.What is the smallest number you must add to 136 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
22. 22.What is the remainder when 232 is divided by 7? (a) 0 (b) 2 (c) 1 (d) 6
23. 23.What is the remainder when 35 is divided by 3? (a) 2 (b) 0 (c) 1
24. 24.What is the units digit of 8^21? (a) 6 (b) 4 (c) 2 (d) 8
25. 25.What is the remainder when 403 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
26. 26.What is the smallest number you must add to 670 to make it divisible by 11? (a) 2 (b) 1 (c) 10 (d) 3
27. 27.Which of these numbers is divisible by 8? (a) 7201 (b) 7200 (c) 7199 (d) 7211
28. 28.What is the remainder when 276 is divided by 7? (a) 2 (b) 4 (c) 5 (d) 3
29. 29.Which of these numbers is divisible by 6? (a) 1727 (b) 1717 (c) 1722 (d) 1731
30. 30.What is the units digit of 2^21? (a) 8 (b) 4 (c) 2 (d) 6
31. 31.Which of these numbers is divisible by 6? (a) 804 (b) 802 (c) 813 (d) 807
32. 32.What is the units digit of 3^54? (a) 3 (b) 7 (c) 1 (d) 9
33. 33.Which of these numbers is divisible by 11? (a) 8292 (b) 8298 (c) 8294 (d) 8315
34. 34.What is the units digit of 3^53? (a) 3 (b) 9 (c) 1 (d) 7
35. 35.Which of these numbers is divisible by 9? (a) 4776 (b) 4784 (c) 4779 (d) 4795
36. 36.What is the remainder when 718 is divided by 10? (a) 2 (b) 8 (c) 9 (d) 7
37. 37.What is the smallest number you must add to 209 to make it divisible by 8? (a) 6 (b) 1 (c) 5 (d) 7
38. 38.Which of these numbers is divisible by 9? (a) 3881 (b) 3874 (c) 3865 (d) 3870
39. 39.What is the units digit of 8^41? (a) 8 (b) 6 (c) 4 (d) 2
40. 40.Which of these numbers is divisible by 3? (a) 1966 (b) 1969 (c) 1972 (d) 1968

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