Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 156 · 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 512 to make it divisible by 9? (a) 3 (b) 8 (c) 2 (d) 1
2. 2.Which of these numbers is divisible by 8? (a) 6484 (b) 6480 (c) 6475 (d) 6490
3. 3.What is the smallest number you must add to 376 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
4. 4.Which of these numbers is divisible by 4? (a) 3584 (b) 3591 (c) 3587 (d) 3581
5. 5.Which of these numbers is divisible by 4? (a) 820 (b) 822 (c) 817 (d) 825
6. 6.What is the smallest number you must add to 799 to make it divisible by 10? (a) 1 (b) 2 (c) 9 (d) 3
7. 7.Which of these numbers is divisible by 4? (a) 1709 (b) 1707 (c) 1703 (d) 1704
8. 8.What is the remainder when 106 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
9. 9.What is the units digit of 3^18? (a) 9 (b) 7 (c) 3 (d) 1
10. 10.Which of these numbers is divisible by 9? (a) 6283 (b) 6279 (c) 6273 (d) 6268
11. 11.Which of these numbers is divisible by 6? (a) 2362 (b) 2351 (c) 2353 (d) 2352
12. 12.What is the remainder when 178 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
13. 13.Which of these numbers is divisible by 3? (a) 653 (b) 650 (c) 646 (d) 648
14. 14.Which of these numbers is divisible by 3? (a) 1931 (b) 1929 (c) 1927 (d) 1934
15. 15.What is the units digit of 2^22? (a) 2 (b) 8 (c) 4 (d) 6
16. 16.What is the units digit of 4^17? (a) 6 (b) 3 (c) 4 (d) 5
17. 17.Which of these numbers is divisible by 6? (a) 4021 (b) 4028 (c) 4017 (d) 4020
18. 18.What is the smallest number you must add to 291 to make it divisible by 4? (a) 1 (b) 3 (c) 2
19. 19.What is the units digit of 9^58? (a) 0 (b) 9 (c) 2 (d) 1
20. 20.Which of these numbers is divisible by 4? (a) 589 (b) 587 (c) 582 (d) 584
21. 21.What is the units digit of 7^23? (a) 9 (b) 7 (c) 1 (d) 3
22. 22.What is the smallest number you must add to 167 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 5
23. 23.Which of these numbers is divisible by 4? (a) 2129 (b) 2135 (c) 2132 (d) 2139
24. 24.What is the units digit of 2^50? (a) 2 (b) 8 (c) 4 (d) 6
25. 25.What is the smallest number you must add to 470 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
26. 26.What is the smallest number you must add to 206 to make it divisible by 3? (a) 1 (b) 2
27. 27.Which of these numbers is divisible by 8? (a) 1245 (b) 1233 (c) 1232 (d) 1231
28. 28.What is the units digit of 9^47? (a) 1 (b) 8 (c) 7 (d) 9
29. 29.What is the remainder when 116 is divided by 3? (a) 1 (b) 2 (c) 0
30. 30.Which of these numbers is divisible by 8? (a) 1461 (b) 1456 (c) 1468 (d) 1452
31. 31.What is the smallest number you must add to 142 to make it divisible by 3? (a) 2 (b) 1
32. 32.What is the smallest number you must add to 117 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
33. 33.What is the smallest number you must add to 230 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 4
34. 34.What is the units digit of 4^40? (a) 7 (b) 6 (c) 4 (d) 5
35. 35.What is the remainder when 138 is divided by 11? (a) 5 (b) 8 (c) 6 (d) 7
36. 36.What is the smallest number you must add to 565 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
37. 37.What is the remainder when 633 is divided by 10? (a) 7 (b) 2 (c) 4 (d) 3
38. 38.What is the smallest number you must add to 317 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
39. 39.What is the remainder when 819 is divided by 11? (a) 5 (b) 7 (c) 6 (d) 4
40. 40.What is the units digit of 2^40? (a) 4 (b) 8 (c) 2 (d) 6

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