Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 748 · 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 remainder when 32 is divided by 3? (a) 2 (b) 1 (c) 0
2. 2.What is the units digit of 4^44? (a) 4 (b) 5 (c) 7 (d) 6
3. 3.What is the smallest number you must add to 94 to make it divisible by 4? (a) 2 (b) 1 (c) 3
4. 4.What is the smallest number you must add to 475 to make it divisible by 7? (a) 1 (b) 6 (c) 2 (d) 3
5. 5.Which of these numbers is divisible by 6? (a) 2528 (b) 2532 (c) 2539 (d) 2534
6. 6.Which of these numbers is divisible by 4? (a) 2730 (b) 2722 (c) 2726 (d) 2724
7. 7.What is the smallest number you must add to 145 to make it divisible by 4? (a) 1 (b) 3 (c) 2
8. 8.What is the remainder when 221 is divided by 5? (a) 2 (b) 0 (c) 1 (d) 4
9. 9.What is the units digit of 4^27? (a) 4 (b) 3 (c) 6 (d) 5
10. 10.Which of these numbers is divisible by 3? (a) 1651 (b) 1655 (c) 1657 (d) 1653
11. 11.What is the remainder when 705 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
12. 12.Which of these numbers is divisible by 6? (a) 1028 (b) 1020 (c) 1018 (d) 1024
13. 13.What is the units digit of 7^31? (a) 3 (b) 7 (c) 9 (d) 1
14. 14.Which of these numbers is divisible by 8? (a) 425 (b) 439 (c) 424 (d) 421
15. 15.What is the smallest number you must add to 239 to make it divisible by 7? (a) 5 (b) 1 (c) 6 (d) 4
16. 16.What is the units digit of 9^28? (a) 1 (b) 9 (c) 2 (d) 0
17. 17.What is the units digit of 9^58? (a) 9 (b) 1 (c) 2 (d) 0
18. 18.What is the smallest number you must add to 199 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
19. 19.What is the remainder when 572 is divided by 10? (a) 2 (b) 8 (c) 1 (d) 3
20. 20.What is the units digit of 3^21? (a) 1 (b) 3 (c) 7 (d) 9
21. 21.Which of these numbers is divisible by 8? (a) 1626 (b) 1639 (c) 1632 (d) 1642
22. 22.What is the smallest number you must add to 129 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 5
23. 23.Which of these numbers is divisible by 9? (a) 1827 (b) 1828 (c) 1823 (d) 1839
24. 24.What is the remainder when 323 is divided by 7? (a) 1 (b) 0 (c) 6 (d) 2
25. 25.Which of these numbers is divisible by 3? (a) 759 (b) 764 (c) 761 (d) 758
26. 26.What is the units digit of 2^15? (a) 2 (b) 8 (c) 6 (d) 4
27. 27.What is the units digit of 8^20? (a) 4 (b) 8 (c) 6 (d) 2
28. 28.What is the smallest number you must add to 149 to make it divisible by 4? (a) 3 (b) 2 (c) 1
29. 29.Which of these numbers is divisible by 4? (a) 2799 (b) 2796 (c) 2802 (d) 2793
30. 30.Which of these numbers is divisible by 9? (a) 5579 (b) 5581 (c) 5595 (d) 5580
31. 31.What is the remainder when 319 is divided by 12? (a) 7 (b) 8 (c) 6 (d) 5
32. 32.What is the units digit of 9^51? (a) 8 (b) 9 (c) 1 (d) 7
33. 33.What is the smallest number you must add to 446 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 4
34. 34.What is the smallest number you must add to 372 to make it divisible by 10? (a) 8 (b) 7 (c) 9 (d) 2
35. 35.Which of these numbers is divisible by 9? (a) 5181 (b) 5161 (c) 5166 (d) 5171
36. 36.Which of these numbers is divisible by 3? (a) 464 (b) 459 (c) 458 (d) 461
37. 37.What is the smallest number you must add to 322 to make it divisible by 4? (a) 1 (b) 2 (c) 3
38. 38.What is the units digit of 2^28? (a) 4 (b) 8 (c) 2 (d) 6
39. 39.What is the smallest number you must add to 139 to make it divisible by 8? (a) 4 (b) 5 (c) 6 (d) 3
40. 40.Which of these numbers is divisible by 8? (a) 5178 (b) 5184 (c) 5196 (d) 5190

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