Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 822 · 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.Which of these numbers is divisible by 3? (a) 625 (b) 631 (c) 627 (d) 628
2. 2.What is the units digit of 2^59? (a) 2 (b) 8 (c) 4 (d) 6
3. 3.What is the smallest number you must add to 232 to make it divisible by 3? (a) 2 (b) 1
4. 4.What is the remainder when 81 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
5. 5.Which of these numbers is divisible by 3? (a) 830 (b) 831 (c) 836 (d) 833
6. 6.What is the smallest number you must add to 447 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 7
7. 7.What is the remainder when 181 is divided by 12? (a) 2 (b) 0 (c) 11 (d) 1
8. 8.What is the smallest number you must add to 98 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
9. 9.What is the units digit of 7^44? (a) 7 (b) 9 (c) 3 (d) 1
10. 10.What is the units digit of 3^19? (a) 1 (b) 3 (c) 7 (d) 9
11. 11.What is the smallest number you must add to 47 to make it divisible by 3? (a) 1 (b) 2
12. 12.What is the remainder when 154 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
13. 13.Which of these numbers is divisible by 11? (a) 7705 (b) 7724 (c) 7720 (d) 7711
14. 14.What is the remainder when 223 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
15. 15.What is the remainder when 161 is divided by 3? (a) 0 (b) 2 (c) 1
16. 16.What is the remainder when 150 is divided by 9? (a) 6 (b) 7 (c) 3 (d) 5
17. 17.Which of these numbers is divisible by 9? (a) 5220 (b) 5219 (c) 5221 (d) 5230
18. 18.Which of these numbers is divisible by 3? (a) 674 (b) 669 (c) 667 (d) 670
19. 19.What is the smallest number you must add to 126 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 6
20. 20.What is the remainder when 56 is divided by 3? (a) 1 (b) 0 (c) 2
21. 21.What is the remainder when 385 is divided by 8? (a) 7 (b) 2 (c) 0 (d) 1
22. 22.What is the smallest number you must add to 247 to make it divisible by 8? (a) 3 (b) 2 (c) 1 (d) 7
23. 23.What is the remainder when 142 is divided by 3? (a) 2 (b) 1 (c) 0
24. 24.What is the remainder when 51 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
25. 25.What is the remainder when 379 is divided by 9? (a) 2 (b) 8 (c) 0 (d) 1
26. 26.Which of these numbers is divisible by 11? (a) 9664 (b) 9656 (c) 9658 (d) 9671
27. 27.Which of these numbers is divisible by 11? (a) 6117 (b) 6114 (c) 6132 (d) 6116
28. 28.What is the units digit of 3^43? (a) 1 (b) 7 (c) 3 (d) 9
29. 29.Which of these numbers is divisible by 9? (a) 4295 (b) 4280 (c) 4291 (d) 4284
30. 30.What is the units digit of 2^32? (a) 2 (b) 8 (c) 4 (d) 6
31. 31.What is the remainder when 547 is divided by 10? (a) 6 (b) 7 (c) 3 (d) 8
32. 32.Which of these numbers is divisible by 6? (a) 3315 (b) 3322 (c) 3312 (d) 3308
33. 33.Which of these numbers is divisible by 6? (a) 5251 (b) 5244 (c) 5240 (d) 5245
34. 34.What is the units digit of 3^29? (a) 1 (b) 7 (c) 9 (d) 3
35. 35.Which of these numbers is divisible by 11? (a) 2660 (b) 2640 (c) 2632 (d) 2642
36. 36.What is the remainder when 570 is divided by 8? (a) 3 (b) 1 (c) 2 (d) 6
37. 37.What is the smallest number you must add to 531 to make it divisible by 11? (a) 8 (b) 9 (c) 3 (d) 7
38. 38.What is the smallest number you must add to 52 to make it divisible by 3? (a) 2 (b) 1
39. 39.Which of these numbers is divisible by 3? (a) 158 (b) 154 (c) 152 (d) 153
40. 40.Which of these numbers is divisible by 11? (a) 1377 (b) 1386 (c) 1406 (d) 1388

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