Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 208 · 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 6? (a) 2356 (b) 2347 (c) 2352 (d) 2359
2. 2.What is the remainder when 249 is divided by 12? (a) 8 (b) 10 (c) 9 (d) 3
3. 3.What is the remainder when 111 is divided by 10? (a) 0 (b) 2 (c) 1 (d) 9
4. 4.What is the smallest number you must add to 525 to make it divisible by 11? (a) 4 (b) 3 (c) 2 (d) 8
5. 5.What is the units digit of 2^53? (a) 4 (b) 6 (c) 8 (d) 2
6. 6.Which of these numbers is divisible by 8? (a) 988 (b) 984 (c) 994 (d) 982
7. 7.What is the remainder when 414 is divided by 10? (a) 3 (b) 5 (c) 6 (d) 4
8. 8.What is the smallest number you must add to 301 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
9. 9.What is the units digit of 8^57? (a) 4 (b) 2 (c) 6 (d) 8
10. 10.What is the remainder when 87 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
11. 11.What is the units digit of 8^34? (a) 8 (b) 4 (c) 6 (d) 2
12. 12.What is the units digit of 9^42? (a) 9 (b) 0 (c) 2 (d) 1
13. 13.What is the units digit of 4^14? (a) 6 (b) 7 (c) 4 (d) 5
14. 14.What is the units digit of 8^22? (a) 6 (b) 2 (c) 8 (d) 4
15. 15.Which of these numbers is divisible by 8? (a) 2881 (b) 2892 (c) 2876 (d) 2880
16. 16.What is the units digit of 9^48? (a) 2 (b) 9 (c) 0 (d) 1
17. 17.What is the remainder when 168 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
18. 18.Which of these numbers is divisible by 4? (a) 2611 (b) 2614 (c) 2617 (d) 2612
19. 19.What is the remainder when 675 is divided by 12? (a) 4 (b) 2 (c) 9 (d) 3
20. 20.What is the smallest number you must add to 275 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
21. 21.What is the remainder when 198 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
22. 22.What is the units digit of 3^30? (a) 3 (b) 7 (c) 9 (d) 1
23. 23.What is the remainder when 208 is divided by 3? (a) 1 (b) 2 (c) 0
24. 24.What is the remainder when 291 is divided by 9? (a) 4 (b) 2 (c) 3 (d) 6
25. 25.Which of these numbers is divisible by 6? (a) 1165 (b) 1173 (c) 1170 (d) 1178
26. 26.What is the smallest number you must add to 99 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
27. 27.What is the smallest number you must add to 795 to make it divisible by 10? (a) 7 (b) 4 (c) 5 (d) 6
28. 28.What is the remainder when 459 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 3
29. 29.Which of these numbers is divisible by 6? (a) 645 (b) 655 (c) 653 (d) 648
30. 30.What is the units digit of 3^57? (a) 1 (b) 3 (c) 7 (d) 9
31. 31.What is the remainder when 367 is divided by 6? (a) 2 (b) 1 (c) 5 (d) 0
32. 32.Which of these numbers is divisible by 4? (a) 1624 (b) 1631 (c) 1627 (d) 1623
33. 33.What is the remainder when 170 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
34. 34.What is the remainder when 181 is divided by 6? (a) 0 (b) 2 (c) 1 (d) 5
35. 35.Which of these numbers is divisible by 8? (a) 3847 (b) 3850 (c) 3839 (d) 3840
36. 36.What is the remainder when 454 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 1
37. 37.What is the units digit of 8^55? (a) 4 (b) 2 (c) 6 (d) 8
38. 38.What is the remainder when 123 is divided by 10? (a) 4 (b) 2 (c) 3 (d) 7
39. 39.What is the remainder when 794 is divided by 12? (a) 3 (b) 2 (c) 1 (d) 10
40. 40.What is the remainder when 197 is divided by 8? (a) 6 (b) 3 (c) 4 (d) 5

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