Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 824 · 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 285 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 5
2. 2.What is the smallest number you must add to 172 to make it divisible by 8? (a) 4 (b) 5 (c) 6 (d) 3
3. 3.What is the smallest number you must add to 533 to make it divisible by 10? (a) 6 (b) 3 (c) 7 (d) 8
4. 4.What is the units digit of 9^30? (a) 0 (b) 2 (c) 1 (d) 9
5. 5.Which of these numbers is divisible by 4? (a) 2533 (b) 2538 (c) 2532 (d) 2530
6. 6.Which of these numbers is divisible by 6? (a) 3008 (b) 2995 (c) 3000 (d) 3003
7. 7.What is the remainder when 482 is divided by 6? (a) 3 (b) 2 (c) 1 (d) 4
8. 8.What is the remainder when 238 is divided by 9? (a) 3 (b) 4 (c) 5 (d) 6
9. 9.What is the remainder when 488 is divided by 9? (a) 7 (b) 2 (c) 3 (d) 1
10. 10.What is the smallest number you must add to 734 to make it divisible by 12? (a) 2 (b) 9 (c) 11 (d) 10
11. 11.What is the smallest number you must add to 929 to make it divisible by 12? (a) 7 (b) 8 (c) 5 (d) 6
12. 12.Which of these numbers is divisible by 3? (a) 2111 (b) 2105 (c) 2107 (d) 2106
13. 13.What is the units digit of 2^37? (a) 6 (b) 4 (c) 8 (d) 2
14. 14.What is the remainder when 763 is divided by 11? (a) 3 (b) 7 (c) 5 (d) 4
15. 15.What is the smallest number you must add to 340 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
16. 16.What is the remainder when 223 is divided by 3? (a) 0 (b) 2 (c) 1
17. 17.What is the units digit of 2^56? (a) 6 (b) 8 (c) 2 (d) 4
18. 18.Which of these numbers is divisible by 3? (a) 1960 (b) 1956 (c) 1955 (d) 1958
19. 19.Which of these numbers is divisible by 11? (a) 6138 (b) 6131 (c) 6139 (d) 6159
20. 20.What is the units digit of 9^24? (a) 0 (b) 1 (c) 9 (d) 2
21. 21.Which of these numbers is divisible by 9? (a) 4072 (b) 4063 (c) 4068 (d) 4082
22. 22.Which of these numbers is divisible by 4? (a) 449 (b) 453 (c) 452 (d) 458
23. 23.What is the remainder when 314 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
24. 24.What is the units digit of 3^46? (a) 7 (b) 9 (c) 3 (d) 1
25. 25.Which of these numbers is divisible by 8? (a) 4969 (b) 4960 (c) 4955 (d) 4963
26. 26.What is the units digit of 9^49? (a) 7 (b) 8 (c) 9 (d) 1
27. 27.Which of these numbers is divisible by 11? (a) 4968 (b) 4961 (c) 4954 (d) 4981
28. 28.Which of these numbers is divisible by 9? (a) 5627 (b) 5618 (c) 5637 (d) 5625
29. 29.What is the smallest number you must add to 220 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 6
30. 30.What is the smallest number you must add to 604 to make it divisible by 11? (a) 2 (b) 3 (c) 1 (d) 10
31. 31.What is the remainder when 532 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 6
32. 32.Which of these numbers is divisible by 4? (a) 1887 (b) 1882 (c) 1889 (d) 1884
33. 33.What is the smallest number you must add to 62 to make it divisible by 3? (a) 2 (b) 1
34. 34.Which of these numbers is divisible by 9? (a) 3611 (b) 3629 (c) 3620 (d) 3618
35. 35.What is the remainder when 829 is divided by 11? (a) 5 (b) 3 (c) 4 (d) 7
36. 36.What is the units digit of 2^59? (a) 2 (b) 8 (c) 4 (d) 6
37. 37.What is the units digit of 9^46? (a) 9 (b) 0 (c) 1 (d) 2
38. 38.What is the smallest number you must add to 631 to make it divisible by 11? (a) 8 (b) 7 (c) 6 (d) 4
39. 39.What is the smallest number you must add to 769 to make it divisible by 10? (a) 9 (b) 2 (c) 3 (d) 1
40. 40.What is the remainder when 201 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 2

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