Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 34 · 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 11? (a) 8603 (b) 8618 (c) 8596 (d) 8602
2. 2.What is the smallest number you must add to 86 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
3. 3.What is the smallest number you must add to 106 to make it divisible by 9? (a) 7 (b) 1 (c) 3 (d) 2
4. 4.Which of these numbers is divisible by 9? (a) 4440 (b) 4448 (c) 4430 (d) 4437
5. 5.What is the units digit of 9^55? (a) 8 (b) 7 (c) 1 (d) 9
6. 6.What is the smallest number you must add to 205 to make it divisible by 3? (a) 1 (b) 2
7. 7.What is the units digit of 9^10? (a) 9 (b) 0 (c) 1 (d) 2
8. 8.What is the smallest number you must add to 178 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 2
9. 9.Which of these numbers is divisible by 9? (a) 7389 (b) 7381 (c) 7393 (d) 7402
10. 10.What is the smallest number you must add to 150 to make it divisible by 8? (a) 1 (b) 2 (c) 6 (d) 3
11. 11.Which of these numbers is divisible by 9? (a) 2328 (b) 2321 (c) 2313 (d) 2312
12. 12.What is the units digit of 8^22? (a) 8 (b) 4 (c) 6 (d) 2
13. 13.Which of these numbers is divisible by 6? (a) 1873 (b) 1878 (c) 1888 (d) 1879
14. 14.Which of these numbers is divisible by 6? (a) 2042 (b) 2030 (c) 2034 (d) 2037
15. 15.What is the smallest number you must add to 860 to make it divisible by 11? (a) 10 (b) 8 (c) 9 (d) 2
16. 16.What is the smallest number you must add to 205 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 1
17. 17.What is the remainder when 344 is divided by 7? (a) 1 (b) 6 (c) 0 (d) 2
18. 18.What is the smallest number you must add to 714 to make it divisible by 11? (a) 1 (b) 2 (c) 3 (d) 10
19. 19.What is the remainder when 342 is divided by 7? (a) 1 (b) 4 (c) 5 (d) 6
20. 20.What is the units digit of 2^48? (a) 8 (b) 6 (c) 4 (d) 2
21. 21.What is the remainder when 415 is divided by 7? (a) 2 (b) 3 (c) 1 (d) 5
22. 22.What is the units digit of 9^57? (a) 7 (b) 1 (c) 9 (d) 8
23. 23.What is the smallest number you must add to 615 to make it divisible by 10? (a) 5 (b) 7 (c) 4 (d) 6
24. 24.What is the units digit of 2^53? (a) 6 (b) 4 (c) 8 (d) 2
25. 25.Which of these numbers is divisible by 8? (a) 5496 (b) 5497 (c) 5508 (d) 5491
26. 26.What is the remainder when 299 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
27. 27.What is the remainder when 221 is divided by 12? (a) 4 (b) 5 (c) 7 (d) 6
28. 28.Which of these numbers is divisible by 3? (a) 1415 (b) 1417 (c) 1412 (d) 1413
29. 29.What is the smallest number you must add to 474 to make it divisible by 10? (a) 4 (b) 5 (c) 6 (d) 7
30. 30.Which of these numbers is divisible by 4? (a) 1963 (b) 1960 (c) 1965 (d) 1957
31. 31.What is the units digit of 8^40? (a) 8 (b) 4 (c) 2 (d) 6
32. 32.What is the remainder when 462 is divided by 12? (a) 8 (b) 5 (c) 6 (d) 7
33. 33.What is the remainder when 72 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
34. 34.What is the smallest number you must add to 134 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
35. 35.What is the units digit of 9^26? (a) 2 (b) 0 (c) 9 (d) 1
36. 36.Which of these numbers is divisible by 3? (a) 803 (b) 804 (c) 809 (d) 805
37. 37.What is the units digit of 8^28? (a) 6 (b) 8 (c) 4 (d) 2
38. 38.What is the remainder when 333 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
39. 39.What is the units digit of 8^60? (a) 8 (b) 6 (c) 4 (d) 2
40. 40.What is the remainder when 370 is divided by 7? (a) 1 (b) 5 (c) 4 (d) 6

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