Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 917 · 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 remainder when 648 is divided by 10? (a) 7 (b) 2 (c) 9 (d) 8
2. 2.What is the smallest number you must add to 239 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
3. 3.What is the units digit of 2^31? (a) 2 (b) 6 (c) 4 (d) 8
4. 4.What is the smallest number you must add to 118 to make it divisible by 3? (a) 2 (b) 1
5. 5.What is the units digit of 2^52? (a) 6 (b) 2 (c) 4 (d) 8
6. 6.Which of these numbers is divisible by 6? (a) 1381 (b) 1380 (c) 1389 (d) 1376
7. 7.What is the remainder when 613 is divided by 10? (a) 7 (b) 2 (c) 4 (d) 3
8. 8.What is the remainder when 85 is divided by 3? (a) 2 (b) 0 (c) 1
9. 9.What is the units digit of 3^23? (a) 3 (b) 7 (c) 9 (d) 1
10. 10.What is the smallest number you must add to 586 to make it divisible by 8? (a) 5 (b) 2 (c) 7 (d) 6
11. 11.What is the remainder when 346 is divided by 10? (a) 6 (b) 7 (c) 4 (d) 5
12. 12.What is the smallest number you must add to 290 to make it divisible by 12? (a) 2 (b) 9 (c) 11 (d) 10
13. 13.Which of these numbers is divisible by 4? (a) 253 (b) 248 (c) 250 (d) 246
14. 14.What is the smallest number you must add to 361 to make it divisible by 9? (a) 7 (b) 8 (c) 6 (d) 1
15. 15.Which of these numbers is divisible by 8? (a) 1712 (b) 1722 (c) 1714 (d) 1705
16. 16.What is the smallest number you must add to 251 to make it divisible by 9? (a) 8 (b) 1 (c) 2 (d) 3
17. 17.Which of these numbers is divisible by 3? (a) 123 (b) 128 (c) 125 (d) 121
18. 18.What is the smallest number you must add to 275 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 2
19. 19.What is the units digit of 8^43? (a) 6 (b) 4 (c) 8 (d) 2
20. 20.What is the remainder when 547 is divided by 12? (a) 7 (b) 8 (c) 5 (d) 6
21. 21.Which of these numbers is divisible by 9? (a) 1418 (b) 1433 (c) 1422 (d) 1429
22. 22.What is the units digit of 8^28? (a) 8 (b) 4 (c) 2 (d) 6
23. 23.What is the smallest number you must add to 484 to make it divisible by 7? (a) 4 (b) 5 (c) 1 (d) 6
24. 24.What is the units digit of 8^27? (a) 2 (b) 6 (c) 4 (d) 8
25. 25.What is the smallest number you must add to 163 to make it divisible by 3? (a) 1 (b) 2
26. 26.Which of these numbers is divisible by 4? (a) 509 (b) 503 (c) 507 (d) 504
27. 27.What is the remainder when 386 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 2
28. 28.What is the smallest number you must add to 70 to make it divisible by 4? (a) 2 (b) 3 (c) 1
29. 29.What is the units digit of 7^23? (a) 1 (b) 7 (c) 3 (d) 9
30. 30.Which of these numbers is divisible by 6? (a) 2253 (b) 2267 (c) 2259 (d) 2256
31. 31.Which of these numbers is divisible by 11? (a) 5407 (b) 5415 (c) 5401 (d) 5395
32. 32.What is the units digit of 8^59? (a) 2 (b) 6 (c) 8 (d) 4
33. 33.Which of these numbers is divisible by 3? (a) 1395 (b) 1399 (c) 1397 (d) 1393
34. 34.Which of these numbers is divisible by 3? (a) 659 (b) 657 (c) 661 (d) 656
35. 35.What is the units digit of 3^59? (a) 9 (b) 3 (c) 7 (d) 1
36. 36.Which of these numbers is divisible by 6? (a) 4605 (b) 4595 (c) 4596 (d) 4597
37. 37.Which of these numbers is divisible by 3? (a) 1588 (b) 1586 (c) 1587 (d) 1592
38. 38.What is the units digit of 7^39? (a) 3 (b) 1 (c) 7 (d) 9
39. 39.What is the remainder when 175 is divided by 6? (a) 0 (b) 1 (c) 5 (d) 2
40. 40.What is the smallest number you must add to 318 to make it divisible by 11? (a) 2 (b) 10 (c) 3 (d) 1

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