Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 129 · 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 161 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
2. 2.What is the smallest number you must add to 195 to make it divisible by 4? (a) 1 (b) 2 (c) 3
3. 3.What is the smallest number you must add to 672 to make it divisible by 11? (a) 8 (b) 1 (c) 10 (d) 9
4. 4.Which of these numbers is divisible by 6? (a) 949 (b) 954 (c) 965 (d) 956
5. 5.What is the remainder when 133 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
6. 6.What is the remainder when 62 is divided by 6? (a) 3 (b) 1 (c) 4 (d) 2
7. 7.What is the remainder when 228 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
8. 8.What is the units digit of 3^14? (a) 3 (b) 7 (c) 1 (d) 9
9. 9.Which of these numbers is divisible by 4? (a) 3521 (b) 3525 (c) 3520 (d) 3518
10. 10.Which of these numbers is divisible by 11? (a) 8061 (b) 8052 (c) 8068 (d) 8050
11. 11.Which of these numbers is divisible by 3? (a) 593 (b) 590 (c) 587 (d) 588
12. 12.Which of these numbers is divisible by 8? (a) 3559 (b) 3546 (c) 3561 (d) 3552
13. 13.Which of these numbers is divisible by 8? (a) 1342 (b) 1336 (c) 1330 (d) 1346
14. 14.What is the remainder when 171 is divided by 12? (a) 4 (b) 9 (c) 2 (d) 3
15. 15.Which of these numbers is divisible by 4? (a) 1793 (b) 1791 (c) 1798 (d) 1792
16. 16.What is the units digit of 4^25? (a) 6 (b) 4 (c) 3 (d) 5
17. 17.What is the remainder when 142 is divided by 3? (a) 1 (b) 2 (c) 0
18. 18.What is the units digit of 2^56? (a) 6 (b) 2 (c) 8 (d) 4
19. 19.Which of these numbers is divisible by 4? (a) 2987 (b) 2988 (c) 2990 (d) 2995
20. 20.What is the units digit of 7^26? (a) 7 (b) 1 (c) 9 (d) 3
21. 21.What is the units digit of 4^44? (a) 7 (b) 4 (c) 6 (d) 5
22. 22.What is the units digit of 4^58? (a) 7 (b) 4 (c) 6 (d) 5
23. 23.What is the remainder when 153 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
24. 24.What is the units digit of 3^59? (a) 1 (b) 7 (c) 9 (d) 3
25. 25.Which of these numbers is divisible by 3? (a) 1396 (b) 1398 (c) 1403 (d) 1399
26. 26.What is the remainder when 617 is divided by 11? (a) 1 (b) 2 (c) 10 (d) 0
27. 27.What is the smallest number you must add to 578 to make it divisible by 10? (a) 3 (b) 8 (c) 1 (d) 2
28. 28.What is the smallest number you must add to 234 to make it divisible by 8? (a) 2 (b) 7 (c) 6 (d) 5
29. 29.Which of these numbers is divisible by 6? (a) 2238 (b) 2237 (c) 2245 (d) 2239
30. 30.Which of these numbers is divisible by 3? (a) 1038 (b) 1036 (c) 1043 (d) 1040
31. 31.What is the units digit of 9^14? (a) 1 (b) 2 (c) 9 (d) 0
32. 32.What is the units digit of 3^34? (a) 7 (b) 3 (c) 1 (d) 9
33. 33.What is the units digit of 7^54? (a) 7 (b) 1 (c) 9 (d) 3
34. 34.What is the smallest number you must add to 102 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
35. 35.What is the smallest number you must add to 71 to make it divisible by 4? (a) 3 (b) 2 (c) 1
36. 36.What is the remainder when 79 is divided by 3? (a) 1 (b) 2 (c) 0
37. 37.What is the units digit of 3^11? (a) 3 (b) 9 (c) 1 (d) 7
38. 38.What is the units digit of 7^16? (a) 9 (b) 3 (c) 7 (d) 1
39. 39.What is the smallest number you must add to 145 to make it divisible by 11? (a) 8 (b) 9 (c) 10 (d) 2
40. 40.Which of these numbers is divisible by 11? (a) 709 (b) 715 (c) 730 (d) 724

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