Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 449 · 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 766 is divided by 12? (a) 9 (b) 2 (c) 11 (d) 10
2. 2.Which of these numbers is divisible by 9? (a) 6498 (b) 6515 (c) 6495 (d) 6502
3. 3.What is the units digit of 9^31? (a) 7 (b) 8 (c) 9 (d) 1
4. 4.What is the smallest number you must add to 639 to make it divisible by 10? (a) 2 (b) 1 (c) 3 (d) 9
5. 5.What is the units digit of 8^32? (a) 6 (b) 8 (c) 2 (d) 4
6. 6.Which of these numbers is divisible by 4? (a) 2961 (b) 2965 (c) 2969 (d) 2964
7. 7.What is the units digit of 2^22? (a) 8 (b) 4 (c) 6 (d) 2
8. 8.What is the remainder when 73 is divided by 3? (a) 1 (b) 2 (c) 0
9. 9.What is the smallest number you must add to 294 to make it divisible by 4? (a) 1 (b) 2 (c) 3
10. 10.What is the smallest number you must add to 391 to make it divisible by 10? (a) 1 (b) 7 (c) 8 (d) 9
11. 11.What is the smallest number you must add to 448 to make it divisible by 9? (a) 7 (b) 3 (c) 2 (d) 1
12. 12.What is the units digit of 9^37? (a) 9 (b) 7 (c) 8 (d) 1
13. 13.What is the remainder when 400 is divided by 7? (a) 2 (b) 0 (c) 1 (d) 6
14. 14.What is the remainder when 122 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2
15. 15.Which of these numbers is divisible by 8? (a) 2000 (b) 1997 (c) 2015 (d) 2007
16. 16.Which of these numbers is divisible by 9? (a) 2558 (b) 2553 (c) 2567 (d) 2556
17. 17.Which of these numbers is divisible by 4? (a) 2257 (b) 2256 (c) 2263 (d) 2254
18. 18.What is the remainder when 167 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
19. 19.What is the remainder when 427 is divided by 8? (a) 5 (b) 3 (c) 4 (d) 2
20. 20.What is the units digit of 8^40? (a) 8 (b) 4 (c) 6 (d) 2
21. 21.What is the remainder when 513 is divided by 8? (a) 2 (b) 0 (c) 7 (d) 1
22. 22.What is the units digit of 4^17? (a) 6 (b) 5 (c) 4 (d) 3
23. 23.Which of these numbers is divisible by 8? (a) 1398 (b) 1400 (c) 1401 (d) 1409
24. 24.What is the smallest number you must add to 166 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
25. 25.What is the remainder when 193 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
26. 26.What is the remainder when 196 is divided by 3? (a) 2 (b) 0 (c) 1
27. 27.Which of these numbers is divisible by 4? (a) 909 (b) 907 (c) 908 (d) 913
28. 28.What is the smallest number you must add to 274 to make it divisible by 10? (a) 7 (b) 5 (c) 4 (d) 6
29. 29.Which of these numbers is divisible by 11? (a) 1128 (b) 1139 (c) 1154 (d) 1133
30. 30.What is the remainder when 107 is divided by 7? (a) 1 (b) 2 (c) 5 (d) 3
31. 31.What is the smallest number you must add to 484 to make it divisible by 7? (a) 5 (b) 1 (c) 6 (d) 4
32. 32.What is the units digit of 3^57? (a) 7 (b) 3 (c) 9 (d) 1
33. 33.What is the units digit of 9^43? (a) 9 (b) 1 (c) 8 (d) 7
34. 34.What is the smallest number you must add to 46 to make it divisible by 4? (a) 3 (b) 2 (c) 1
35. 35.What is the smallest number you must add to 230 to make it divisible by 3? (a) 1 (b) 2
36. 36.What is the remainder when 61 is divided by 6? (a) 2 (b) 0 (c) 5 (d) 1
37. 37.What is the smallest number you must add to 103 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
38. 38.Which of these numbers is divisible by 3? (a) 875 (b) 877 (c) 871 (d) 873
39. 39.What is the remainder when 83 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
40. 40.Which of these numbers is divisible by 3? (a) 1852 (b) 1848 (c) 1850 (d) 1846

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