Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 767 · 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 208 to make it divisible by 3? (a) 2 (b) 1
2. 2.What is the remainder when 527 is divided by 8? (a) 1 (b) 6 (c) 5 (d) 7
3. 3.What is the units digit of 2^47? (a) 8 (b) 2 (c) 4 (d) 6
4. 4.What is the smallest number you must add to 109 to make it divisible by 4? (a) 2 (b) 1 (c) 3
5. 5.Which of these numbers is divisible by 3? (a) 749 (b) 744 (c) 745 (d) 743
6. 6.What is the remainder when 876 is divided by 11? (a) 4 (b) 6 (c) 8 (d) 7
7. 7.What is the smallest number you must add to 181 to make it divisible by 9? (a) 1 (b) 6 (c) 7 (d) 8
8. 8.Which of these numbers is divisible by 3? (a) 2141 (b) 2146 (c) 2143 (d) 2142
9. 9.What is the remainder when 79 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
10. 10.What is the units digit of 8^60? (a) 8 (b) 4 (c) 2 (d) 6
11. 11.What is the smallest number you must add to 599 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 7
12. 12.Which of these numbers is divisible by 11? (a) 7682 (b) 7668 (c) 7690 (d) 7678
13. 13.What is the smallest number you must add to 271 to make it divisible by 4? (a) 2 (b) 1 (c) 3
14. 14.What is the remainder when 677 is divided by 12? (a) 7 (b) 4 (c) 5 (d) 6
15. 15.What is the units digit of 9^45? (a) 1 (b) 8 (c) 7 (d) 9
16. 16.Which of these numbers is divisible by 9? (a) 1061 (b) 1062 (c) 1075 (d) 1069
17. 17.What is the units digit of 4^27? (a) 6 (b) 5 (c) 4 (d) 3
18. 18.What is the smallest number you must add to 116 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
19. 19.Which of these numbers is divisible by 11? (a) 4383 (b) 4366 (c) 4370 (d) 4367
20. 20.What is the remainder when 62 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
21. 21.What is the remainder when 569 is divided by 8? (a) 7 (b) 0 (c) 1 (d) 2
22. 22.What is the units digit of 3^26? (a) 1 (b) 7 (c) 3 (d) 9
23. 23.What is the units digit of 3^41? (a) 3 (b) 7 (c) 9 (d) 1
24. 24.What is the smallest number you must add to 126 to make it divisible by 11? (a) 6 (b) 7 (c) 8 (d) 5
25. 25.What is the remainder when 402 is divided by 10? (a) 8 (b) 3 (c) 2 (d) 1
26. 26.What is the remainder when 591 is divided by 9? (a) 6 (b) 3 (c) 5 (d) 7
27. 27.Which of these numbers is divisible by 3? (a) 1600 (b) 1602 (c) 1604 (d) 1607
28. 28.What is the units digit of 3^22? (a) 9 (b) 7 (c) 1 (d) 3
29. 29.What is the remainder when 286 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
30. 30.Which of these numbers is divisible by 3? (a) 310 (b) 312 (c) 314 (d) 317
31. 31.What is the units digit of 3^25? (a) 9 (b) 3 (c) 1 (d) 7
32. 32.What is the remainder when 256 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
33. 33.What is the smallest number you must add to 171 to make it divisible by 4? (a) 2 (b) 3 (c) 1
34. 34.What is the smallest number you must add to 121 to make it divisible by 9? (a) 6 (b) 4 (c) 7 (d) 5
35. 35.What is the units digit of 3^19? (a) 9 (b) 7 (c) 1 (d) 3
36. 36.What is the smallest number you must add to 558 to make it divisible by 7? (a) 2 (b) 5 (c) 3 (d) 1
37. 37.Which of these numbers is divisible by 9? (a) 3607 (b) 3615 (c) 3600 (d) 3592
38. 38.Which of these numbers is divisible by 11? (a) 7506 (b) 7489 (c) 7491 (d) 7495
39. 39.What is the units digit of 9^28? (a) 0 (b) 9 (c) 2 (d) 1
40. 40.Which of these numbers is divisible by 11? (a) 2579 (b) 2570 (c) 2586 (d) 2574

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