Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 609 · 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 287 to make it divisible by 4? (a) 1 (b) 3 (c) 2
2. 2.What is the remainder when 163 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
3. 3.What is the smallest number you must add to 292 to make it divisible by 7? (a) 1 (b) 5 (c) 3 (d) 2
4. 4.Which of these numbers is divisible by 4? (a) 553 (b) 556 (c) 563 (d) 559
5. 5.What is the remainder when 153 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
6. 6.What is the remainder when 109 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
7. 7.Which of these numbers is divisible by 6? (a) 4950 (b) 4949 (c) 4961 (d) 4953
8. 8.What is the smallest number you must add to 524 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
9. 9.What is the units digit of 9^10? (a) 0 (b) 2 (c) 1 (d) 9
10. 10.What is the units digit of 3^36? (a) 7 (b) 3 (c) 9 (d) 1
11. 11.What is the units digit of 3^50? (a) 7 (b) 3 (c) 9 (d) 1
12. 12.What is the units digit of 2^27? (a) 4 (b) 8 (c) 6 (d) 2
13. 13.Which of these numbers is divisible by 6? (a) 4176 (b) 4185 (c) 4175 (d) 4177
14. 14.Which of these numbers is divisible by 9? (a) 2260 (b) 2250 (c) 2255 (d) 2245
15. 15.What is the units digit of 8^30? (a) 6 (b) 2 (c) 8 (d) 4
16. 16.What is the remainder when 225 is divided by 11? (a) 6 (b) 4 (c) 7 (d) 5
17. 17.Which of these numbers is divisible by 8? (a) 3970 (b) 3982 (c) 3976 (d) 3990
18. 18.What is the remainder when 179 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 5
19. 19.What is the units digit of 3^14? (a) 1 (b) 7 (c) 3 (d) 9
20. 20.What is the smallest number you must add to 333 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 5
21. 21.Which of these numbers is divisible by 4? (a) 864 (b) 870 (c) 863 (d) 865
22. 22.What is the smallest number you must add to 168 to make it divisible by 10? (a) 1 (b) 2 (c) 3 (d) 8
23. 23.Which of these numbers is divisible by 9? (a) 7415 (b) 7430 (c) 7416 (d) 7417
24. 24.What is the units digit of 2^44? (a) 8 (b) 6 (c) 4 (d) 2
25. 25.What is the smallest number you must add to 76 to make it divisible by 3? (a) 2 (b) 1
26. 26.Which of these numbers is divisible by 6? (a) 805 (b) 803 (c) 813 (d) 804
27. 27.What is the remainder when 618 is divided by 8? (a) 1 (b) 6 (c) 3 (d) 2
28. 28.What is the units digit of 8^41? (a) 6 (b) 4 (c) 2 (d) 8
29. 29.What is the remainder when 157 is divided by 3? (a) 2 (b) 0 (c) 1
30. 30.Which of these numbers is divisible by 8? (a) 4182 (b) 4176 (c) 4172 (d) 4186
31. 31.What is the remainder when 43 is divided by 3? (a) 0 (b) 2 (c) 1
32. 32.What is the units digit of 9^21? (a) 8 (b) 7 (c) 1 (d) 9
33. 33.What is the units digit of 3^23? (a) 9 (b) 1 (c) 3 (d) 7
34. 34.What is the smallest number you must add to 248 to make it divisible by 12? (a) 8 (b) 5 (c) 4 (d) 3
35. 35.What is the remainder when 187 is divided by 10? (a) 8 (b) 7 (c) 3 (d) 6
36. 36.What is the remainder when 318 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
37. 37.What is the units digit of 2^48? (a) 4 (b) 6 (c) 2 (d) 8
38. 38.What is the smallest number you must add to 309 to make it divisible by 4? (a) 3 (b) 1 (c) 2
39. 39.What is the smallest number you must add to 248 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 4
40. 40.What is the smallest number you must add to 173 to make it divisible by 3? (a) 1 (b) 2

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