Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 855 · 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 4^36? (a) 7 (b) 5 (c) 6 (d) 4
2. 2.What is the units digit of 9^42? (a) 9 (b) 1 (c) 0 (d) 2
3. 3.Which of these numbers is divisible by 4? (a) 3295 (b) 3289 (c) 3297 (d) 3292
4. 4.What is the smallest number you must add to 115 to make it divisible by 3? (a) 1 (b) 2
5. 5.Which of these numbers is divisible by 8? (a) 5070 (b) 5064 (c) 5060 (d) 5074
6. 6.What is the remainder when 781 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
7. 7.What is the smallest number you must add to 293 to make it divisible by 7? (a) 1 (b) 2 (c) 3 (d) 6
8. 8.What is the smallest number you must add to 677 to make it divisible by 10? (a) 4 (b) 7 (c) 3 (d) 2
9. 9.Which of these numbers is divisible by 6? (a) 2921 (b) 2908 (c) 2911 (d) 2910
10. 10.Which of these numbers is divisible by 3? (a) 425 (b) 426 (c) 428 (d) 430
11. 11.What is the units digit of 7^33? (a) 3 (b) 7 (c) 9 (d) 1
12. 12.What is the units digit of 3^15? (a) 3 (b) 7 (c) 9 (d) 1
13. 13.Which of these numbers is divisible by 4? (a) 2691 (b) 2684 (c) 2682 (d) 2686
14. 14.Which of these numbers is divisible by 3? (a) 1929 (b) 1930 (c) 1927 (d) 1934
15. 15.What is the units digit of 9^40? (a) 9 (b) 2 (c) 1 (d) 0
16. 16.Which of these numbers is divisible by 3? (a) 1112 (b) 1107 (c) 1108 (d) 1105
17. 17.What is the units digit of 4^56? (a) 4 (b) 6 (c) 7 (d) 5
18. 18.What is the units digit of 4^40? (a) 5 (b) 4 (c) 7 (d) 6
19. 19.Which of these numbers is divisible by 3? (a) 1217 (b) 1213 (c) 1210 (d) 1212
20. 20.Which of these numbers is divisible by 9? (a) 7978 (b) 7983 (c) 7987 (d) 7993
21. 21.What is the remainder when 276 is divided by 8? (a) 6 (b) 3 (c) 5 (d) 4
22. 22.What is the remainder when 724 is divided by 11? (a) 2 (b) 9 (c) 10 (d) 8
23. 23.What is the units digit of 9^14? (a) 2 (b) 0 (c) 9 (d) 1
24. 24.What is the smallest number you must add to 317 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
25. 25.Which of these numbers is divisible by 6? (a) 2641 (b) 2634 (c) 2635 (d) 2631
26. 26.Which of these numbers is divisible by 11? (a) 3367 (b) 3366 (c) 3360 (d) 3385
27. 27.What is the units digit of 2^42? (a) 2 (b) 6 (c) 4 (d) 8
28. 28.What is the smallest number you must add to 83 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
29. 29.What is the units digit of 2^43? (a) 6 (b) 8 (c) 4 (d) 2
30. 30.What is the units digit of 2^18? (a) 6 (b) 8 (c) 4 (d) 2
31. 31.Which of these numbers is divisible by 8? (a) 1729 (b) 1727 (c) 1728 (d) 1742
32. 32.What is the units digit of 4^14? (a) 4 (b) 7 (c) 5 (d) 6
33. 33.Which of these numbers is divisible by 9? (a) 5094 (b) 5098 (c) 5086 (d) 5111
34. 34.What is the units digit of 8^53? (a) 4 (b) 2 (c) 8 (d) 6
35. 35.Which of these numbers is divisible by 3? (a) 950 (b) 947 (c) 948 (d) 952
36. 36.Which of these numbers is divisible by 9? (a) 2106 (b) 2105 (c) 2121 (d) 2111
37. 37.What is the remainder when 234 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
38. 38.Which of these numbers is divisible by 4? (a) 846 (b) 839 (c) 840 (d) 843
39. 39.Which of these numbers is divisible by 11? (a) 4873 (b) 4871 (c) 4883 (d) 4886
40. 40.What is the remainder when 223 is divided by 12? (a) 5 (b) 8 (c) 7 (d) 6

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