Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 67 · 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 3^44? (a) 7 (b) 1 (c) 3 (d) 9
2. 2.What is the units digit of 4^14? (a) 4 (b) 5 (c) 7 (d) 6
3. 3.What is the remainder when 518 is divided by 8? (a) 5 (b) 2 (c) 6 (d) 7
4. 4.What is the units digit of 9^18? (a) 9 (b) 0 (c) 2 (d) 1
5. 5.Which of these numbers is divisible by 11? (a) 9679 (b) 9652 (c) 9658 (d) 9664
6. 6.Which of these numbers is divisible by 9? (a) 6243 (b) 6256 (c) 6246 (d) 6254
7. 7.What is the remainder when 695 is divided by 12? (a) 9 (b) 1 (c) 10 (d) 11
8. 8.What is the smallest number you must add to 217 to make it divisible by 10? (a) 2 (b) 4 (c) 7 (d) 3
9. 9.What is the remainder when 357 is divided by 11? (a) 7 (b) 4 (c) 6 (d) 5
10. 10.What is the units digit of 9^11? (a) 8 (b) 7 (c) 1 (d) 9
11. 11.What is the smallest number you must add to 127 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
12. 12.What is the units digit of 4^16? (a) 5 (b) 4 (c) 6 (d) 7
13. 13.What is the smallest number you must add to 218 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
14. 14.Which of these numbers is divisible by 8? (a) 5885 (b) 5891 (c) 5888 (d) 5902
15. 15.What is the units digit of 9^59? (a) 1 (b) 7 (c) 8 (d) 9
16. 16.What is the units digit of 3^23? (a) 1 (b) 7 (c) 3 (d) 9
17. 17.What is the smallest number you must add to 164 to make it divisible by 3? (a) 1 (b) 2
18. 18.What is the units digit of 9^31? (a) 7 (b) 9 (c) 8 (d) 1
19. 19.What is the remainder when 697 is divided by 10? (a) 6 (b) 8 (c) 3 (d) 7
20. 20.Which of these numbers is divisible by 4? (a) 3416 (b) 3415 (c) 3419 (d) 3423
21. 21.Which of these numbers is divisible by 3? (a) 2663 (b) 2669 (c) 2666 (d) 2664
22. 22.Which of these numbers is divisible by 4? (a) 934 (b) 928 (c) 931 (d) 926
23. 23.What is the remainder when 625 is divided by 12? (a) 2 (b) 11 (c) 0 (d) 1
24. 24.Which of these numbers is divisible by 6? (a) 740 (b) 738 (c) 745 (d) 736
25. 25.What is the smallest number you must add to 362 to make it divisible by 11? (a) 10 (b) 3 (c) 1 (d) 2
26. 26.What is the remainder when 150 is divided by 8? (a) 2 (b) 5 (c) 6 (d) 7
27. 27.Which of these numbers is divisible by 4? (a) 1979 (b) 1983 (c) 1974 (d) 1976
28. 28.What is the remainder when 329 is divided by 6? (a) 1 (b) 4 (c) 5 (d) 3
29. 29.What is the remainder when 556 is divided by 12? (a) 5 (b) 8 (c) 4 (d) 3
30. 30.What is the smallest number you must add to 389 to make it divisible by 7? (a) 5 (b) 2 (c) 4 (d) 3
31. 31.What is the units digit of 4^41? (a) 5 (b) 3 (c) 6 (d) 4
32. 32.Which of these numbers is divisible by 4? (a) 3534 (b) 3531 (c) 3528 (d) 3525
33. 33.Which of these numbers is divisible by 3? (a) 1312 (b) 1306 (c) 1308 (d) 1310
34. 34.What is the units digit of 3^26? (a) 1 (b) 9 (c) 7 (d) 3
35. 35.What is the units digit of 8^18? (a) 2 (b) 8 (c) 4 (d) 6
36. 36.Which of these numbers is divisible by 11? (a) 2061 (b) 2049 (c) 2045 (d) 2046
37. 37.What is the smallest number you must add to 137 to make it divisible by 4? (a) 1 (b) 3 (c) 2
38. 38.What is the smallest number you must add to 52 to make it divisible by 3? (a) 1 (b) 2
39. 39.What is the remainder when 414 is divided by 12? (a) 5 (b) 7 (c) 6 (d) 8
40. 40.What is the remainder when 582 is divided by 8? (a) 2 (b) 6 (c) 5 (d) 7

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