Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 217 · 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 194 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
2. 2.What is the units digit of 9^32? (a) 9 (b) 2 (c) 1 (d) 0
3. 3.What is the remainder when 648 is divided by 10? (a) 2 (b) 8 (c) 9 (d) 7
4. 4.What is the remainder when 182 is divided by 3? (a) 1 (b) 0 (c) 2
5. 5.What is the remainder when 194 is divided by 7? (a) 2 (b) 4 (c) 6 (d) 5
6. 6.What is the units digit of 4^37? (a) 6 (b) 4 (c) 3 (d) 5
7. 7.What is the smallest number you must add to 310 to make it divisible by 11? (a) 8 (b) 9 (c) 2 (d) 10
8. 8.Which of these numbers is divisible by 9? (a) 3708 (b) 3702 (c) 3710 (d) 3720
9. 9.What is the smallest number you must add to 70 to make it divisible by 4? (a) 1 (b) 2 (c) 3
10. 10.What is the units digit of 3^52? (a) 7 (b) 3 (c) 1 (d) 9
11. 11.What is the smallest number you must add to 442 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 1
12. 12.Which of these numbers is divisible by 9? (a) 5584 (b) 5579 (c) 5580 (d) 5591
13. 13.What is the remainder when 708 is divided by 11? (a) 7 (b) 5 (c) 3 (d) 4
14. 14.What is the remainder when 559 is divided by 9? (a) 8 (b) 0 (c) 2 (d) 1
15. 15.What is the units digit of 8^28? (a) 4 (b) 6 (c) 8 (d) 2
16. 16.What is the smallest number you must add to 334 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 4
17. 17.What is the smallest number you must add to 281 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
18. 18.Which of these numbers is divisible by 3? (a) 503 (b) 498 (c) 497 (d) 500
19. 19.Which of these numbers is divisible by 4? (a) 2185 (b) 2177 (c) 2180 (d) 2182
20. 20.What is the units digit of 9^57? (a) 7 (b) 9 (c) 8 (d) 1
21. 21.What is the smallest number you must add to 230 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 6
22. 22.What is the remainder when 260 is divided by 12? (a) 9 (b) 4 (c) 7 (d) 8
23. 23.Which of these numbers is divisible by 3? (a) 322 (b) 321 (c) 320 (d) 325
24. 24.Which of these numbers is divisible by 9? (a) 5918 (b) 5911 (c) 5898 (d) 5904
25. 25.Which of these numbers is divisible by 4? (a) 1827 (b) 1823 (c) 1824 (d) 1830
26. 26.What is the smallest number you must add to 219 to make it divisible by 10? (a) 2 (b) 3 (c) 9 (d) 1
27. 27.What is the units digit of 3^23? (a) 1 (b) 9 (c) 7 (d) 3
28. 28.What is the units digit of 7^46? (a) 3 (b) 9 (c) 7 (d) 1
29. 29.Which of these numbers is divisible by 6? (a) 2136 (b) 2135 (c) 2137 (d) 2146
30. 30.Which of these numbers is divisible by 11? (a) 3049 (b) 3036 (c) 3029 (d) 3038
31. 31.What is the units digit of 2^39? (a) 4 (b) 6 (c) 8 (d) 2
32. 32.What is the remainder when 182 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
33. 33.What is the units digit of 7^48? (a) 7 (b) 1 (c) 9 (d) 3
34. 34.Which of these numbers is divisible by 6? (a) 4751 (b) 4746 (c) 4743 (d) 4755
35. 35.What is the remainder when 411 is divided by 12? (a) 3 (b) 4 (c) 2 (d) 9
36. 36.Which of these numbers is divisible by 11? (a) 7797 (b) 7819 (c) 7801 (d) 7799
37. 37.What is the units digit of 8^23? (a) 2 (b) 8 (c) 6 (d) 4
38. 38.What is the remainder when 213 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
39. 39.What is the remainder when 183 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
40. 40.Which of these numbers is divisible by 8? (a) 4808 (b) 4823 (c) 4810 (d) 4805

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