Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 407 · 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 751 to make it divisible by 11? (a) 7 (b) 8 (c) 9 (d) 3
2. 2.What is the smallest number you must add to 424 to make it divisible by 9? (a) 1 (b) 6 (c) 8 (d) 7
3. 3.What is the remainder when 168 is divided by 9? (a) 5 (b) 3 (c) 7 (d) 6
4. 4.What is the units digit of 3^48? (a) 1 (b) 9 (c) 7 (d) 3
5. 5.What is the units digit of 3^15? (a) 1 (b) 3 (c) 9 (d) 7
6. 6.What is the smallest number you must add to 135 to make it divisible by 12? (a) 10 (b) 3 (c) 9 (d) 8
7. 7.What is the units digit of 9^20? (a) 0 (b) 2 (c) 1 (d) 9
8. 8.What is the remainder when 136 is divided by 11? (a) 5 (b) 3 (c) 7 (d) 4
9. 9.What is the remainder when 185 is divided by 6? (a) 4 (b) 1 (c) 5 (d) 3
10. 10.Which of these numbers is divisible by 4? (a) 1097 (b) 1094 (c) 1096 (d) 1103
11. 11.What is the units digit of 7^12? (a) 1 (b) 7 (c) 9 (d) 3
12. 12.Which of these numbers is divisible by 4? (a) 2611 (b) 2602 (c) 2605 (d) 2604
13. 13.What is the remainder when 142 is divided by 10? (a) 3 (b) 8 (c) 1 (d) 2
14. 14.Which of these numbers is divisible by 6? (a) 5114 (b) 5118 (c) 5120 (d) 5126
15. 15.What is the units digit of 7^33? (a) 1 (b) 3 (c) 9 (d) 7
16. 16.What is the units digit of 3^58? (a) 7 (b) 3 (c) 9 (d) 1
17. 17.What is the remainder when 352 is divided by 7? (a) 3 (b) 2 (c) 5 (d) 1
18. 18.What is the units digit of 3^44? (a) 1 (b) 3 (c) 7 (d) 9
19. 19.What is the units digit of 4^52? (a) 5 (b) 7 (c) 4 (d) 6
20. 20.What is the remainder when 80 is divided by 3? (a) 1 (b) 2 (c) 0
21. 21.What is the units digit of 9^21? (a) 7 (b) 1 (c) 9 (d) 8
22. 22.What is the units digit of 3^21? (a) 1 (b) 9 (c) 7 (d) 3
23. 23.What is the smallest number you must add to 782 to make it divisible by 12? (a) 10 (b) 11 (c) 9 (d) 2
24. 24.What is the remainder when 191 is divided by 3? (a) 0 (b) 1 (c) 2
25. 25.What is the smallest number you must add to 248 to make it divisible by 10? (a) 8 (b) 3 (c) 1 (d) 2
26. 26.What is the smallest number you must add to 194 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 5
27. 27.What is the units digit of 8^34? (a) 6 (b) 4 (c) 2 (d) 8
28. 28.What is the units digit of 9^32? (a) 9 (b) 2 (c) 1 (d) 0
29. 29.What is the remainder when 789 is divided by 10? (a) 7 (b) 1 (c) 9 (d) 8
30. 30.What is the remainder when 315 is divided by 12? (a) 9 (b) 3 (c) 4 (d) 2
31. 31.Which of these numbers is divisible by 9? (a) 6267 (b) 6277 (c) 6264 (d) 6262
32. 32.What is the smallest number you must add to 208 to make it divisible by 9? (a) 6 (b) 1 (c) 7 (d) 8
33. 33.What is the smallest number you must add to 277 to make it divisible by 12? (a) 10 (b) 9 (c) 1 (d) 11
34. 34.What is the remainder when 258 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
35. 35.What is the units digit of 7^54? (a) 7 (b) 1 (c) 9 (d) 3
36. 36.What is the smallest number you must add to 117 to make it divisible by 4? (a) 1 (b) 2 (c) 3
37. 37.Which of these numbers is divisible by 6? (a) 5053 (b) 5050 (c) 5042 (d) 5046
38. 38.What is the remainder when 197 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
39. 39.What is the remainder when 409 is divided by 7? (a) 5 (b) 4 (c) 2 (d) 3
40. 40.What is the smallest number you must add to 141 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5

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