Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 198 · 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 9? (a) 7294 (b) 7300 (c) 7290 (d) 7288
2. 2.Which of these numbers is divisible by 11? (a) 4401 (b) 4400 (c) 4393 (d) 4412
3. 3.What is the smallest number you must add to 155 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the smallest number you must add to 230 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
5. 5.What is the units digit of 8^53? (a) 2 (b) 6 (c) 4 (d) 8
6. 6.Which of these numbers is divisible by 4? (a) 1581 (b) 1577 (c) 1576 (d) 1575
7. 7.What is the units digit of 3^55? (a) 7 (b) 3 (c) 1 (d) 9
8. 8.What is the units digit of 2^21? (a) 2 (b) 8 (c) 6 (d) 4
9. 9.What is the remainder when 305 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
10. 10.What is the units digit of 8^51? (a) 4 (b) 2 (c) 6 (d) 8
11. 11.Which of these numbers is divisible by 8? (a) 871 (b) 857 (c) 856 (d) 854
12. 12.What is the units digit of 7^43? (a) 7 (b) 3 (c) 9 (d) 1
13. 13.What is the remainder when 278 is divided by 6? (a) 3 (b) 1 (c) 2 (d) 4
14. 14.What is the smallest number you must add to 368 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
15. 15.What is the smallest number you must add to 247 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 5
16. 16.What is the units digit of 9^38? (a) 2 (b) 1 (c) 0 (d) 9
17. 17.What is the units digit of 8^40? (a) 8 (b) 6 (c) 2 (d) 4
18. 18.Which of these numbers is divisible by 11? (a) 5379 (b) 5377 (c) 5395 (d) 5380
19. 19.What is the units digit of 2^15? (a) 8 (b) 6 (c) 4 (d) 2
20. 20.What is the remainder when 149 is divided by 10? (a) 8 (b) 7 (c) 1 (d) 9
21. 21.What is the remainder when 50 is divided by 3? (a) 2 (b) 1 (c) 0
22. 22.What is the remainder when 910 is divided by 12? (a) 2 (b) 9 (c) 10 (d) 11
23. 23.What is the units digit of 2^58? (a) 6 (b) 2 (c) 8 (d) 4
24. 24.What is the smallest number you must add to 314 to make it divisible by 12? (a) 2 (b) 9 (c) 11 (d) 10
25. 25.What is the smallest number you must add to 54 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
26. 26.What is the units digit of 2^50? (a) 4 (b) 6 (c) 8 (d) 2
27. 27.What is the units digit of 2^43? (a) 4 (b) 6 (c) 2 (d) 8
28. 28.What is the smallest number you must add to 406 to make it divisible by 12? (a) 10 (b) 2 (c) 1 (d) 3
29. 29.Which of these numbers is divisible by 9? (a) 5299 (b) 5286 (c) 5282 (d) 5283
30. 30.What is the remainder when 635 is divided by 9? (a) 4 (b) 6 (c) 5 (d) 7
31. 31.Which of these numbers is divisible by 11? (a) 9670 (b) 9681 (c) 9663 (d) 9669
32. 32.What is the smallest number you must add to 184 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
33. 33.What is the units digit of 4^14? (a) 7 (b) 6 (c) 4 (d) 5
34. 34.What is the units digit of 2^22? (a) 6 (b) 4 (c) 8 (d) 2
35. 35.What is the units digit of 7^25? (a) 3 (b) 1 (c) 7 (d) 9
36. 36.What is the smallest number you must add to 64 to make it divisible by 3? (a) 2 (b) 1
37. 37.Which of these numbers is divisible by 4? (a) 3154 (b) 3148 (c) 3147 (d) 3150
38. 38.What is the smallest number you must add to 278 to make it divisible by 9? (a) 8 (b) 1 (c) 3 (d) 2
39. 39.Which of these numbers is divisible by 6? (a) 1598 (b) 1610 (c) 1603 (d) 1602
40. 40.What is the units digit of 2^18? (a) 2 (b) 4 (c) 8 (d) 6

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