Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 116 · 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 430 to make it divisible by 6? (a) 2 (b) 4 (c) 1 (d) 3
2. 2.Which of these numbers is divisible by 3? (a) 187 (b) 190 (c) 189 (d) 194
3. 3.What is the smallest number you must add to 399 to make it divisible by 8? (a) 2 (b) 3 (c) 7 (d) 1
4. 4.What is the remainder when 436 is divided by 6? (a) 3 (b) 2 (c) 5 (d) 4
5. 5.Which of these numbers is divisible by 8? (a) 6435 (b) 6431 (c) 6424 (d) 6417
6. 6.Which of these numbers is divisible by 4? (a) 716 (b) 713 (c) 717 (d) 721
7. 7.What is the units digit of 2^26? (a) 2 (b) 8 (c) 4 (d) 6
8. 8.What is the smallest number you must add to 106 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
9. 9.What is the remainder when 301 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
10. 10.What is the remainder when 700 is divided by 9? (a) 6 (b) 2 (c) 8 (d) 7
11. 11.What is the units digit of 2^58? (a) 6 (b) 4 (c) 8 (d) 2
12. 12.What is the smallest number you must add to 271 to make it divisible by 4? (a) 2 (b) 1 (c) 3
13. 13.What is the smallest number you must add to 860 to make it divisible by 11? (a) 10 (b) 2 (c) 9 (d) 8
14. 14.What is the units digit of 2^31? (a) 4 (b) 8 (c) 2 (d) 6
15. 15.What is the units digit of 3^34? (a) 9 (b) 3 (c) 1 (d) 7
16. 16.What is the remainder when 619 is divided by 11? (a) 2 (b) 3 (c) 4 (d) 8
17. 17.Which of these numbers is divisible by 4? (a) 1263 (b) 1267 (c) 1269 (d) 1264
18. 18.Which of these numbers is divisible by 6? (a) 4354 (b) 4348 (c) 4339 (d) 4344
19. 19.What is the remainder when 150 is divided by 9? (a) 3 (b) 7 (c) 6 (d) 5
20. 20.What is the units digit of 9^32? (a) 2 (b) 9 (c) 0 (d) 1
21. 21.What is the smallest number you must add to 187 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
22. 22.What is the smallest number you must add to 727 to make it divisible by 9? (a) 7 (b) 2 (c) 3 (d) 1
23. 23.What is the units digit of 2^30? (a) 8 (b) 6 (c) 2 (d) 4
24. 24.What is the units digit of 9^55? (a) 1 (b) 8 (c) 9 (d) 7
25. 25.What is the smallest number you must add to 229 to make it divisible by 6? (a) 3 (b) 1 (c) 5 (d) 4
26. 26.What is the remainder when 211 is divided by 11? (a) 1 (b) 9 (c) 2 (d) 3
27. 27.What is the smallest number you must add to 133 to make it divisible by 11? (a) 8 (b) 10 (c) 1 (d) 9
28. 28.What is the units digit of 8^44? (a) 2 (b) 8 (c) 6 (d) 4
29. 29.What is the smallest number you must add to 685 to make it divisible by 10? (a) 6 (b) 5 (c) 7 (d) 4
30. 30.Which of these numbers is divisible by 6? (a) 5294 (b) 5301 (c) 5308 (d) 5298
31. 31.Which of these numbers is divisible by 11? (a) 2519 (b) 2520 (c) 2514 (d) 2538
32. 32.Which of these numbers is divisible by 11? (a) 2970 (b) 2973 (c) 2984 (d) 2960
33. 33.What is the remainder when 481 is divided by 8? (a) 2 (b) 7 (c) 1 (d) 0
34. 34.Which of these numbers is divisible by 8? (a) 627 (b) 611 (c) 618 (d) 616
35. 35.What is the units digit of 2^17? (a) 2 (b) 6 (c) 8 (d) 4
36. 36.What is the smallest number you must add to 94 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
37. 37.What is the smallest number you must add to 465 to make it divisible by 9? (a) 6 (b) 4 (c) 3 (d) 2
38. 38.What is the remainder when 139 is divided by 3? (a) 0 (b) 2 (c) 1
39. 39.Which of these numbers is divisible by 8? (a) 2696 (b) 2711 (c) 2694 (d) 2698
40. 40.What is the smallest number you must add to 402 to make it divisible by 9? (a) 4 (b) 2 (c) 3 (d) 6

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