Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 695 · 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 3^59? (a) 1 (b) 7 (c) 9 (d) 3
2. 2.What is the smallest number you must add to 77 to make it divisible by 4? (a) 1 (b) 2 (c) 3
3. 3.What is the smallest number you must add to 430 to make it divisible by 7? (a) 4 (b) 5 (c) 3 (d) 6
4. 4.What is the units digit of 7^18? (a) 9 (b) 3 (c) 7 (d) 1
5. 5.What is the remainder when 140 is divided by 9? (a) 4 (b) 6 (c) 7 (d) 5
6. 6.What is the smallest number you must add to 392 to make it divisible by 11? (a) 3 (b) 7 (c) 5 (d) 4
7. 7.What is the smallest number you must add to 134 to make it divisible by 4? (a) 1 (b) 2 (c) 3
8. 8.Which of these numbers is divisible by 4? (a) 311 (b) 305 (c) 304 (d) 301
9. 9.What is the units digit of 7^16? (a) 7 (b) 9 (c) 3 (d) 1
10. 10.Which of these numbers is divisible by 8? (a) 2784 (b) 2783 (c) 2798 (d) 2785
11. 11.What is the smallest number you must add to 553 to make it divisible by 10? (a) 3 (b) 8 (c) 7 (d) 6
12. 12.Which of these numbers is divisible by 9? (a) 4339 (b) 4349 (c) 4362 (d) 4347
13. 13.What is the remainder when 133 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
14. 14.What is the units digit of 4^11? (a) 6 (b) 3 (c) 4 (d) 5
15. 15.Which of these numbers is divisible by 4? (a) 1425 (b) 1423 (c) 1424 (d) 1431
16. 16.Which of these numbers is divisible by 3? (a) 414 (b) 412 (c) 416 (d) 418
17. 17.Which of these numbers is divisible by 8? (a) 3466 (b) 3456 (c) 3457 (d) 3451
18. 18.What is the remainder when 364 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 6
19. 19.What is the units digit of 2^24? (a) 4 (b) 8 (c) 6 (d) 2
20. 20.Which of these numbers is divisible by 4? (a) 659 (b) 666 (c) 660 (d) 663
21. 21.What is the units digit of 9^44? (a) 2 (b) 9 (c) 0 (d) 1
22. 22.What is the units digit of 4^17? (a) 5 (b) 6 (c) 3 (d) 4
23. 23.What is the remainder when 274 is divided by 9? (a) 3 (b) 4 (c) 6 (d) 5
24. 24.What is the smallest number you must add to 53 to make it divisible by 3? (a) 2 (b) 1
25. 25.What is the remainder when 736 is divided by 10? (a) 6 (b) 4 (c) 7 (d) 5
26. 26.What is the units digit of 2^39? (a) 4 (b) 8 (c) 6 (d) 2
27. 27.What is the remainder when 586 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
28. 28.Which of these numbers is divisible by 9? (a) 2219 (b) 2203 (c) 2205 (d) 2211
29. 29.What is the units digit of 4^12? (a) 4 (b) 7 (c) 5 (d) 6
30. 30.What is the smallest number you must add to 550 to make it divisible by 7? (a) 5 (b) 4 (c) 2 (d) 3
31. 31.Which of these numbers is divisible by 3? (a) 1533 (b) 1535 (c) 1538 (d) 1531
32. 32.What is the smallest number you must add to 338 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
33. 33.Which of these numbers is divisible by 8? (a) 3812 (b) 3820 (c) 3808 (d) 3805
34. 34.What is the units digit of 3^30? (a) 9 (b) 3 (c) 1 (d) 7
35. 35.What is the remainder when 313 is divided by 6? (a) 1 (b) 2 (c) 5 (d) 0
36. 36.Which of these numbers is divisible by 3? (a) 1449 (b) 1453 (c) 1450 (d) 1448
37. 37.What is the units digit of 3^27? (a) 9 (b) 7 (c) 3 (d) 1
38. 38.What is the remainder when 132 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
39. 39.Which of these numbers is divisible by 3? (a) 135 (b) 137 (c) 139 (d) 133
40. 40.What is the remainder when 219 is divided by 7? (a) 5 (b) 1 (c) 3 (d) 2

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