Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 763 · 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 598 to make it divisible by 10? (a) 1 (b) 8 (c) 3 (d) 2
2. 2.What is the smallest number you must add to 191 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 2
3. 3.What is the smallest number you must add to 50 to make it divisible by 4? (a) 3 (b) 2 (c) 1
4. 4.What is the smallest number you must add to 222 to make it divisible by 9? (a) 3 (b) 6 (c) 2 (d) 4
5. 5.Which of these numbers is divisible by 11? (a) 8087 (b) 8085 (c) 8081 (d) 8106
6. 6.Which of these numbers is divisible by 3? (a) 2182 (b) 2181 (c) 2185 (d) 2180
7. 7.Which of these numbers is divisible by 3? (a) 2682 (b) 2686 (c) 2681 (d) 2684
8. 8.What is the units digit of 8^18? (a) 2 (b) 6 (c) 8 (d) 4
9. 9.Which of these numbers is divisible by 8? (a) 684 (b) 672 (c) 666 (d) 679
10. 10.What is the smallest number you must add to 614 to make it divisible by 11? (a) 1 (b) 2 (c) 3 (d) 9
11. 11.What is the units digit of 2^24? (a) 2 (b) 8 (c) 4 (d) 6
12. 12.What is the smallest number you must add to 46 to make it divisible by 4? (a) 2 (b) 3 (c) 1
13. 13.Which of these numbers is divisible by 3? (a) 1651 (b) 1646 (c) 1649 (d) 1647
14. 14.What is the remainder when 95 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 5
15. 15.What is the remainder when 663 is divided by 10? (a) 4 (b) 7 (c) 3 (d) 2
16. 16.What is the units digit of 3^13? (a) 7 (b) 1 (c) 9 (d) 3
17. 17.What is the units digit of 7^32? (a) 3 (b) 7 (c) 9 (d) 1
18. 18.Which of these numbers is divisible by 3? (a) 731 (b) 727 (c) 726 (d) 725
19. 19.What is the units digit of 7^44? (a) 9 (b) 3 (c) 7 (d) 1
20. 20.What is the smallest number you must add to 166 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
21. 21.What is the smallest number you must add to 76 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
22. 22.What is the units digit of 2^10? (a) 4 (b) 8 (c) 2 (d) 6
23. 23.What is the units digit of 9^22? (a) 1 (b) 2 (c) 0 (d) 9
24. 24.What is the smallest number you must add to 163 to make it divisible by 4? (a) 1 (b) 3 (c) 2
25. 25.Which of these numbers is divisible by 3? (a) 1521 (b) 1520 (c) 1526 (d) 1522
26. 26.What is the units digit of 9^28? (a) 0 (b) 2 (c) 9 (d) 1
27. 27.What is the units digit of 8^58? (a) 6 (b) 8 (c) 2 (d) 4
28. 28.What is the smallest number you must add to 42 to make it divisible by 4? (a) 2 (b) 3 (c) 1
29. 29.What is the smallest number you must add to 54 to make it divisible by 4? (a) 3 (b) 1 (c) 2
30. 30.Which of these numbers is divisible by 4? (a) 3087 (b) 3095 (c) 3089 (d) 3088
31. 31.Which of these numbers is divisible by 6? (a) 2650 (b) 2655 (c) 2652 (d) 2663
32. 32.What is the units digit of 8^40? (a) 2 (b) 8 (c) 4 (d) 6
33. 33.What is the remainder when 71 is divided by 3? (a) 2 (b) 1 (c) 0
34. 34.Which of these numbers is divisible by 6? (a) 1525 (b) 1523 (c) 1517 (d) 1518
35. 35.What is the smallest number you must add to 365 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 1
36. 36.What is the units digit of 2^60? (a) 4 (b) 8 (c) 2 (d) 6
37. 37.Which of these numbers is divisible by 4? (a) 2878 (b) 2874 (c) 2876 (d) 2883
38. 38.What is the remainder when 113 is divided by 9? (a) 7 (b) 4 (c) 5 (d) 6
39. 39.What is the smallest number you must add to 309 to make it divisible by 4? (a) 2 (b) 3 (c) 1
40. 40.What is the units digit of 3^31? (a) 3 (b) 9 (c) 7 (d) 1

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