Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 774 · 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 499 is divided by 11? (a) 7 (b) 3 (c) 5 (d) 4
2. 2.What is the remainder when 131 is divided by 3? (a) 0 (b) 1 (c) 2
3. 3.What is the smallest number you must add to 632 to make it divisible by 10? (a) 2 (b) 8 (c) 9 (d) 7
4. 4.What is the units digit of 9^17? (a) 7 (b) 1 (c) 9 (d) 8
5. 5.What is the remainder when 279 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
6. 6.What is the remainder when 791 is divided by 10? (a) 1 (b) 9 (c) 2 (d) 0
7. 7.What is the units digit of 2^42? (a) 4 (b) 6 (c) 8 (d) 2
8. 8.What is the smallest number you must add to 201 to make it divisible by 8? (a) 6 (b) 1 (c) 7 (d) 5
9. 9.What is the units digit of 2^10? (a) 8 (b) 6 (c) 2 (d) 4
10. 10.What is the smallest number you must add to 148 to make it divisible by 3? (a) 1 (b) 2
11. 11.What is the smallest number you must add to 72 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
12. 12.What is the remainder when 414 is divided by 7? (a) 2 (b) 6 (c) 0 (d) 1
13. 13.What is the smallest number you must add to 223 to make it divisible by 4? (a) 3 (b) 2 (c) 1
14. 14.What is the remainder when 442 is divided by 11? (a) 1 (b) 9 (c) 3 (d) 2
15. 15.What is the smallest number you must add to 220 to make it divisible by 7? (a) 6 (b) 4 (c) 3 (d) 5
16. 16.What is the remainder when 124 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
17. 17.What is the units digit of 9^11? (a) 8 (b) 9 (c) 1 (d) 7
18. 18.What is the remainder when 419 is divided by 11? (a) 1 (b) 0 (c) 2 (d) 10
19. 19.Which of these numbers is divisible by 8? (a) 4689 (b) 4696 (c) 4705 (d) 4700
20. 20.What is the smallest number you must add to 178 to make it divisible by 7? (a) 6 (b) 3 (c) 5 (d) 4
21. 21.Which of these numbers is divisible by 9? (a) 8086 (b) 8073 (c) 8080 (d) 8069
22. 22.What is the units digit of 8^33? (a) 6 (b) 2 (c) 4 (d) 8
23. 23.What is the units digit of 9^19? (a) 1 (b) 8 (c) 7 (d) 9
24. 24.What is the smallest number you must add to 158 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
25. 25.What is the units digit of 9^30? (a) 9 (b) 0 (c) 2 (d) 1
26. 26.What is the remainder when 91 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
27. 27.Which of these numbers is divisible by 6? (a) 3733 (b) 3740 (c) 3732 (d) 3731
28. 28.What is the smallest number you must add to 113 to make it divisible by 11? (a) 8 (b) 3 (c) 9 (d) 7
29. 29.What is the remainder when 234 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
30. 30.What is the remainder when 683 is divided by 9? (a) 7 (b) 6 (c) 1 (d) 8
31. 31.What is the remainder when 676 is divided by 11? (a) 7 (b) 6 (c) 5 (d) 4
32. 32.Which of these numbers is divisible by 4? (a) 2758 (b) 2762 (c) 2753 (d) 2756
33. 33.Which of these numbers is divisible by 3? (a) 364 (b) 371 (c) 366 (d) 368
34. 34.What is the units digit of 3^10? (a) 7 (b) 3 (c) 1 (d) 9
35. 35.Which of these numbers is divisible by 6? (a) 3707 (b) 3702 (c) 3709 (d) 3699
36. 36.Which of these numbers is divisible by 9? (a) 5048 (b) 5049 (c) 5060 (d) 5051
37. 37.Which of these numbers is divisible by 11? (a) 6227 (b) 6226 (c) 6216 (d) 6242
38. 38.Which of these numbers is divisible by 4? (a) 1701 (b) 1711 (c) 1705 (d) 1704
39. 39.What is the units digit of 3^19? (a) 7 (b) 3 (c) 1 (d) 9
40. 40.What is the units digit of 3^24? (a) 3 (b) 7 (c) 1 (d) 9

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