Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 920 · 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 257 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 1
2. 2.Which of these numbers is divisible by 6? (a) 336 (b) 335 (c) 345 (d) 339
3. 3.What is the remainder when 131 is divided by 3? (a) 2 (b) 0 (c) 1
4. 4.Which of these numbers is divisible by 6? (a) 2611 (b) 2604 (c) 2602 (d) 2606
5. 5.What is the smallest number you must add to 455 to make it divisible by 9? (a) 3 (b) 5 (c) 6 (d) 4
6. 6.What is the remainder when 460 is divided by 9? (a) 0 (b) 2 (c) 8 (d) 1
7. 7.What is the units digit of 4^19? (a) 3 (b) 6 (c) 5 (d) 4
8. 8.Which of these numbers is divisible by 9? (a) 6307 (b) 6313 (c) 6296 (d) 6300
9. 9.What is the units digit of 7^56? (a) 9 (b) 3 (c) 1 (d) 7
10. 10.What is the units digit of 4^41? (a) 5 (b) 6 (c) 3 (d) 4
11. 11.What is the remainder when 197 is divided by 3? (a) 2 (b) 0 (c) 1
12. 12.What is the units digit of 2^28? (a) 2 (b) 4 (c) 8 (d) 6
13. 13.What is the remainder when 56 is divided by 3? (a) 1 (b) 2 (c) 0
14. 14.What is the units digit of 4^44? (a) 4 (b) 6 (c) 5 (d) 7
15. 15.What is the smallest number you must add to 257 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
16. 16.What is the smallest number you must add to 303 to make it divisible by 8? (a) 7 (b) 3 (c) 2 (d) 1
17. 17.What is the units digit of 3^31? (a) 3 (b) 9 (c) 7 (d) 1
18. 18.What is the remainder when 406 is divided by 12? (a) 9 (b) 2 (c) 10 (d) 11
19. 19.Which of these numbers is divisible by 3? (a) 1979 (b) 1976 (c) 1972 (d) 1974
20. 20.What is the units digit of 2^58? (a) 6 (b) 8 (c) 2 (d) 4
21. 21.Which of these numbers is divisible by 8? (a) 2648 (b) 2655 (c) 2641 (d) 2663
22. 22.What is the units digit of 2^20? (a) 4 (b) 6 (c) 2 (d) 8
23. 23.What is the units digit of 2^57? (a) 2 (b) 8 (c) 6 (d) 4
24. 24.What is the units digit of 9^34? (a) 0 (b) 1 (c) 9 (d) 2
25. 25.What is the remainder when 461 is divided by 7? (a) 6 (b) 5 (c) 1 (d) 4
26. 26.What is the smallest number you must add to 189 to make it divisible by 10? (a) 9 (b) 3 (c) 2 (d) 1
27. 27.Which of these numbers is divisible by 6? (a) 2802 (b) 2797 (c) 2809 (d) 2803
28. 28.What is the remainder when 89 is divided by 8? (a) 0 (b) 2 (c) 1 (d) 7
29. 29.What is the remainder when 194 is divided by 3? (a) 0 (b) 1 (c) 2
30. 30.What is the smallest number you must add to 63 to make it divisible by 4? (a) 2 (b) 1 (c) 3
31. 31.What is the smallest number you must add to 310 to make it divisible by 11? (a) 2 (b) 9 (c) 8 (d) 10
32. 32.Which of these numbers is divisible by 8? (a) 3303 (b) 3281 (c) 3289 (d) 3288
33. 33.What is the remainder when 853 is divided by 12? (a) 11 (b) 1 (c) 0 (d) 2
34. 34.What is the units digit of 3^20? (a) 9 (b) 7 (c) 3 (d) 1
35. 35.What is the remainder when 632 is divided by 10? (a) 3 (b) 8 (c) 2 (d) 1
36. 36.What is the units digit of 3^34? (a) 3 (b) 1 (c) 7 (d) 9
37. 37.What is the smallest number you must add to 357 to make it divisible by 11? (a) 6 (b) 7 (c) 8 (d) 5
38. 38.What is the remainder when 630 is divided by 8? (a) 2 (b) 7 (c) 6 (d) 5
39. 39.What is the smallest number you must add to 376 to make it divisible by 11? (a) 2 (b) 10 (c) 8 (d) 9
40. 40.What is the remainder when 222 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2

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