Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 971 · 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 smallest number you must add to 497 to make it divisible by 11? (a) 9 (b) 8 (c) 2 (d) 10
2. 2.Which of these numbers is divisible by 3? (a) 1642 (b) 1640 (c) 1641 (d) 1646
3. 3.What is the remainder when 106 is divided by 5? (a) 2 (b) 1 (c) 0 (d) 4
4. 4.What is the units digit of 2^20? (a) 4 (b) 2 (c) 6 (d) 8
5. 5.What is the units digit of 4^14? (a) 7 (b) 5 (c) 4 (d) 6
6. 6.Which of these numbers is divisible by 4? (a) 3293 (b) 3296 (c) 3302 (d) 3298
7. 7.Which of these numbers is divisible by 8? (a) 1044 (b) 1060 (c) 1048 (d) 1053
8. 8.What is the remainder when 446 is divided by 10? (a) 5 (b) 4 (c) 7 (d) 6
9. 9.What is the smallest number you must add to 43 to make it divisible by 4? (a) 3 (b) 1 (c) 2
10. 10.What is the smallest number you must add to 167 to make it divisible by 9? (a) 3 (b) 5 (c) 6 (d) 4
11. 11.What is the remainder when 127 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
12. 12.Which of these numbers is divisible by 6? (a) 1654 (b) 1645 (c) 1650 (d) 1658
13. 13.What is the smallest number you must add to 80 to make it divisible by 3? (a) 2 (b) 1
14. 14.Which of these numbers is divisible by 3? (a) 726 (b) 730 (c) 728 (d) 724
15. 15.What is the remainder when 127 is divided by 8? (a) 7 (b) 1 (c) 5 (d) 6
16. 16.What is the units digit of 8^10? (a) 8 (b) 6 (c) 4 (d) 2
17. 17.What is the smallest number you must add to 297 to make it divisible by 4? (a) 3 (b) 1 (c) 2
18. 18.What is the smallest number you must add to 86 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
19. 19.What is the smallest number you must add to 585 to make it divisible by 8? (a) 1 (b) 6 (c) 5 (d) 7
20. 20.What is the smallest number you must add to 560 to make it divisible by 9? (a) 2 (b) 7 (c) 8 (d) 6
21. 21.What is the remainder when 421 is divided by 7? (a) 2 (b) 1 (c) 6 (d) 0
22. 22.What is the units digit of 9^50? (a) 2 (b) 0 (c) 1 (d) 9
23. 23.What is the remainder when 429 is divided by 12? (a) 10 (b) 9 (c) 8 (d) 3
24. 24.What is the smallest number you must add to 103 to make it divisible by 4? (a) 1 (b) 2 (c) 3
25. 25.What is the smallest number you must add to 773 to make it divisible by 10? (a) 8 (b) 7 (c) 3 (d) 6
26. 26.Which of these numbers is divisible by 4? (a) 230 (b) 235 (c) 227 (d) 228
27. 27.Which of these numbers is divisible by 4? (a) 1566 (b) 1570 (c) 1564 (d) 1562
28. 28.What is the remainder when 249 is divided by 9? (a) 7 (b) 5 (c) 6 (d) 3
29. 29.What is the remainder when 425 is divided by 6? (a) 1 (b) 5 (c) 3 (d) 4
30. 30.What is the units digit of 3^23? (a) 1 (b) 9 (c) 3 (d) 7
31. 31.What is the smallest number you must add to 601 to make it divisible by 11? (a) 4 (b) 3 (c) 5 (d) 7
32. 32.Which of these numbers is divisible by 8? (a) 7178 (b) 7185 (c) 7193 (d) 7184
33. 33.What is the remainder when 552 is divided by 9? (a) 3 (b) 2 (c) 4 (d) 6
34. 34.What is the remainder when 200 is divided by 3? (a) 0 (b) 1 (c) 2
35. 35.What is the units digit of 9^22? (a) 2 (b) 0 (c) 1 (d) 9
36. 36.Which of these numbers is divisible by 8? (a) 5471 (b) 5456 (c) 5452 (d) 5457
37. 37.What is the smallest number you must add to 719 to make it divisible by 11? (a) 8 (b) 7 (c) 4 (d) 6
38. 38.Which of these numbers is divisible by 9? (a) 2556 (b) 2567 (c) 2551 (d) 2559
39. 39.Which of these numbers is divisible by 3? (a) 807 (b) 812 (c) 805 (d) 809
40. 40.Which of these numbers is divisible by 6? (a) 1737 (b) 1743 (c) 1740 (d) 1750

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