Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 527 · 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 3^36? (a) 7 (b) 1 (c) 3 (d) 9
2. 2.Which of these numbers is divisible by 8? (a) 1511 (b) 1512 (c) 1523 (d) 1514
3. 3.What is the smallest number you must add to 68 to make it divisible by 3? (a) 2 (b) 1
4. 4.What is the remainder when 323 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
5. 5.Which of these numbers is divisible by 8? (a) 3624 (b) 3620 (c) 3638 (d) 3626
6. 6.What is the units digit of 8^38? (a) 8 (b) 2 (c) 6 (d) 4
7. 7.What is the remainder when 218 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
8. 8.What is the remainder when 237 is divided by 12? (a) 8 (b) 3 (c) 10 (d) 9
9. 9.What is the units digit of 7^47? (a) 7 (b) 9 (c) 1 (d) 3
10. 10.What is the smallest number you must add to 91 to make it divisible by 4? (a) 3 (b) 2 (c) 1
11. 11.What is the remainder when 80 is divided by 3? (a) 1 (b) 0 (c) 2
12. 12.What is the remainder when 381 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 5
13. 13.What is the smallest number you must add to 294 to make it divisible by 10? (a) 5 (b) 7 (c) 6 (d) 4
14. 14.What is the units digit of 8^11? (a) 2 (b) 8 (c) 4 (d) 6
15. 15.Which of these numbers is divisible by 8? (a) 1472 (b) 1483 (c) 1476 (d) 1467
16. 16.What is the remainder when 446 is divided by 10? (a) 6 (b) 7 (c) 5 (d) 4
17. 17.What is the remainder when 173 is divided by 11? (a) 3 (b) 9 (c) 8 (d) 7
18. 18.What is the smallest number you must add to 64 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
19. 19.What is the smallest number you must add to 122 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3
20. 20.What is the smallest number you must add to 140 to make it divisible by 3? (a) 1 (b) 2
21. 21.What is the units digit of 9^27? (a) 1 (b) 7 (c) 9 (d) 8
22. 22.Which of these numbers is divisible by 11? (a) 1672 (b) 1666 (c) 1685 (d) 1673
23. 23.What is the smallest number you must add to 680 to make it divisible by 11? (a) 3 (b) 1 (c) 2 (d) 9
24. 24.What is the remainder when 151 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
25. 25.What is the units digit of 8^40? (a) 2 (b) 4 (c) 6 (d) 8
26. 26.What is the remainder when 143 is divided by 8? (a) 1 (b) 5 (c) 6 (d) 7
27. 27.Which of these numbers is divisible by 11? (a) 8269 (b) 8254 (c) 8250 (d) 8245
28. 28.Which of these numbers is divisible by 9? (a) 2761 (b) 2767 (c) 2776 (d) 2763
29. 29.What is the units digit of 2^50? (a) 8 (b) 4 (c) 2 (d) 6
30. 30.Which of these numbers is divisible by 8? (a) 2896 (b) 2897 (c) 2906 (d) 2891
31. 31.What is the smallest number you must add to 333 to make it divisible by 12? (a) 4 (b) 2 (c) 9 (d) 3
32. 32.What is the smallest number you must add to 764 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
33. 33.What is the remainder when 119 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
34. 34.What is the units digit of 4^14? (a) 7 (b) 5 (c) 6 (d) 4
35. 35.What is the remainder when 866 is divided by 11? (a) 7 (b) 8 (c) 3 (d) 9
36. 36.What is the units digit of 4^46? (a) 7 (b) 6 (c) 4 (d) 5
37. 37.What is the remainder when 326 is divided by 7? (a) 5 (b) 6 (c) 3 (d) 4
38. 38.What is the units digit of 7^15? (a) 7 (b) 3 (c) 1 (d) 9
39. 39.What is the smallest number you must add to 246 to make it divisible by 4? (a) 1 (b) 3 (c) 2
40. 40.Which of these numbers is divisible by 9? (a) 6520 (b) 6542 (c) 6525 (d) 6533

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