Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 164 · 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 units digit of 9^13? (a) 8 (b) 9 (c) 1 (d) 7
2. 2.Which of these numbers is divisible by 9? (a) 3107 (b) 3120 (c) 3129 (d) 3114
3. 3.Which of these numbers is divisible by 4? (a) 627 (b) 623 (c) 624 (d) 631
4. 4.What is the smallest number you must add to 246 to make it divisible by 11? (a) 7 (b) 8 (c) 6 (d) 4
5. 5.What is the smallest number you must add to 288 to make it divisible by 10? (a) 8 (b) 3 (c) 1 (d) 2
6. 6.What is the units digit of 3^23? (a) 7 (b) 9 (c) 1 (d) 3
7. 7.What is the units digit of 8^36? (a) 2 (b) 8 (c) 6 (d) 4
8. 8.What is the remainder when 530 is divided by 9? (a) 7 (b) 8 (c) 6 (d) 1
9. 9.Which of these numbers is divisible by 11? (a) 7170 (b) 7189 (c) 7172 (d) 7174
10. 10.What is the smallest number you must add to 433 to make it divisible by 12? (a) 9 (b) 10 (c) 1 (d) 11
11. 11.What is the smallest number you must add to 400 to make it divisible by 11? (a) 6 (b) 8 (c) 4 (d) 7
12. 12.What is the smallest number you must add to 718 to make it divisible by 9? (a) 3 (b) 7 (c) 2 (d) 1
13. 13.What is the units digit of 9^31? (a) 1 (b) 7 (c) 8 (d) 9
14. 14.What is the units digit of 3^26? (a) 7 (b) 9 (c) 1 (d) 3
15. 15.What is the units digit of 8^17? (a) 8 (b) 6 (c) 2 (d) 4
16. 16.What is the remainder when 299 is divided by 10? (a) 1 (b) 8 (c) 7 (d) 9
17. 17.What is the units digit of 8^50? (a) 8 (b) 6 (c) 4 (d) 2
18. 18.What is the remainder when 153 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
19. 19.What is the smallest number you must add to 303 to make it divisible by 4? (a) 1 (b) 2 (c) 3
20. 20.What is the smallest number you must add to 688 to make it divisible by 9? (a) 7 (b) 6 (c) 5 (d) 4
21. 21.Which of these numbers is divisible by 11? (a) 5143 (b) 5150 (c) 5148 (d) 5160
22. 22.Which of these numbers is divisible by 9? (a) 2107 (b) 2118 (c) 2103 (d) 2106
23. 23.What is the smallest number you must add to 252 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
24. 24.What is the smallest number you must add to 189 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
25. 25.Which of these numbers is divisible by 3? (a) 842 (b) 845 (c) 843 (d) 847
26. 26.Which of these numbers is divisible by 3? (a) 1116 (b) 1117 (c) 1120 (d) 1114
27. 27.What is the remainder when 124 is divided by 3? (a) 1 (b) 0 (c) 2
28. 28.What is the remainder when 121 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
29. 29.What is the units digit of 4^55? (a) 3 (b) 5 (c) 6 (d) 4
30. 30.What is the remainder when 50 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
31. 31.Which of these numbers is divisible by 3? (a) 650 (b) 645 (c) 646 (d) 644
32. 32.Which of these numbers is divisible by 3? (a) 1834 (b) 1838 (c) 1833 (d) 1832
33. 33.What is the units digit of 4^42? (a) 6 (b) 7 (c) 4 (d) 5
34. 34.What is the units digit of 2^42? (a) 8 (b) 2 (c) 6 (d) 4
35. 35.What is the units digit of 8^22? (a) 4 (b) 2 (c) 6 (d) 8
36. 36.What is the smallest number you must add to 420 to make it divisible by 9? (a) 6 (b) 4 (c) 3 (d) 2
37. 37.What is the remainder when 95 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
38. 38.What is the remainder when 613 is divided by 12? (a) 1 (b) 0 (c) 2 (d) 11
39. 39.What is the units digit of 4^37? (a) 4 (b) 6 (c) 5 (d) 3
40. 40.What is the smallest number you must add to 114 to make it divisible by 4? (a) 1 (b) 3 (c) 2

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