Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 915 · 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 279 is divided by 12? (a) 2 (b) 3 (c) 9 (d) 4
2. 2.What is the units digit of 7^53? (a) 1 (b) 3 (c) 9 (d) 7
3. 3.What is the remainder when 768 is divided by 10? (a) 7 (b) 8 (c) 2 (d) 9
4. 4.What is the smallest number you must add to 286 to make it divisible by 6? (a) 2 (b) 4 (c) 1 (d) 3
5. 5.Which of these numbers is divisible by 3? (a) 188 (b) 190 (c) 185 (d) 186
6. 6.What is the remainder when 103 is divided by 3? (a) 0 (b) 1 (c) 2
7. 7.Which of these numbers is divisible by 8? (a) 6658 (b) 6654 (c) 6668 (d) 6656
8. 8.Which of these numbers is divisible by 4? (a) 3104 (b) 3109 (c) 3105 (d) 3102
9. 9.What is the smallest number you must add to 91 to make it divisible by 8? (a) 4 (b) 6 (c) 3 (d) 5
10. 10.What is the smallest number you must add to 810 to make it divisible by 11? (a) 7 (b) 4 (c) 3 (d) 5
11. 11.What is the remainder when 348 is divided by 11? (a) 4 (b) 8 (c) 7 (d) 6
12. 12.Which of these numbers is divisible by 11? (a) 938 (b) 930 (c) 923 (d) 924
13. 13.What is the smallest number you must add to 323 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
14. 14.What is the remainder when 143 is divided by 8? (a) 6 (b) 7 (c) 1 (d) 5
15. 15.Which of these numbers is divisible by 6? (a) 1109 (b) 1120 (c) 1112 (d) 1110
16. 16.Which of these numbers is divisible by 4? (a) 3056 (b) 3062 (c) 3058 (d) 3053
17. 17.What is the smallest number you must add to 266 to make it divisible by 4? (a) 2 (b) 3 (c) 1
18. 18.Which of these numbers is divisible by 6? (a) 5127 (b) 5118 (c) 5121 (d) 5114
19. 19.What is the remainder when 595 is divided by 11? (a) 10 (b) 0 (c) 1 (d) 2
20. 20.Which of these numbers is divisible by 4? (a) 2294 (b) 2290 (c) 2299 (d) 2292
21. 21.What is the remainder when 496 is divided by 9? (a) 0 (b) 2 (c) 1 (d) 8
22. 22.What is the units digit of 9^15? (a) 8 (b) 7 (c) 1 (d) 9
23. 23.What is the smallest number you must add to 696 to make it divisible by 10? (a) 3 (b) 4 (c) 5 (d) 6
24. 24.What is the smallest number you must add to 315 to make it divisible by 12? (a) 9 (b) 8 (c) 3 (d) 10
25. 25.What is the units digit of 4^17? (a) 3 (b) 4 (c) 6 (d) 5
26. 26.What is the units digit of 2^17? (a) 8 (b) 2 (c) 4 (d) 6
27. 27.What is the smallest number you must add to 166 to make it divisible by 6? (a) 2 (b) 4 (c) 1 (d) 3
28. 28.What is the smallest number you must add to 165 to make it divisible by 12? (a) 9 (b) 4 (c) 3 (d) 2
29. 29.Which of these numbers is divisible by 3? (a) 356 (b) 353 (c) 351 (d) 350
30. 30.What is the smallest number you must add to 313 to make it divisible by 4? (a) 1 (b) 3 (c) 2
31. 31.What is the units digit of 8^17? (a) 4 (b) 6 (c) 2 (d) 8
32. 32.What is the remainder when 170 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 1
33. 33.Which of these numbers is divisible by 4? (a) 1442 (b) 1446 (c) 1450 (d) 1444
34. 34.What is the units digit of 2^16? (a) 6 (b) 4 (c) 2 (d) 8
35. 35.What is the remainder when 175 is divided by 9? (a) 4 (b) 5 (c) 6 (d) 3
36. 36.What is the units digit of 8^49? (a) 4 (b) 2 (c) 6 (d) 8
37. 37.What is the units digit of 3^48? (a) 3 (b) 9 (c) 7 (d) 1
38. 38.Which of these numbers is divisible by 8? (a) 2024 (b) 2034 (c) 2026 (d) 2022
39. 39.What is the units digit of 3^44? (a) 3 (b) 7 (c) 9 (d) 1
40. 40.What is the units digit of 7^39? (a) 9 (b) 7 (c) 3 (d) 1

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