Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 754 · 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 262 to make it divisible by 4? (a) 1 (b) 3 (c) 2
2. 2.Which of these numbers is divisible by 6? (a) 1052 (b) 1056 (c) 1065 (d) 1059
3. 3.Which of these numbers is divisible by 4? (a) 647 (b) 642 (c) 644 (d) 649
4. 4.What is the remainder when 219 is divided by 10? (a) 7 (b) 8 (c) 9 (d) 1
5. 5.Which of these numbers is divisible by 9? (a) 5455 (b) 5445 (c) 5450 (d) 5443
6. 6.What is the smallest number you must add to 542 to make it divisible by 9? (a) 2 (b) 6 (c) 8 (d) 7
7. 7.What is the remainder when 156 is divided by 10? (a) 5 (b) 7 (c) 4 (d) 6
8. 8.What is the smallest number you must add to 211 to make it divisible by 4? (a) 3 (b) 2 (c) 1
9. 9.Which of these numbers is divisible by 4? (a) 2287 (b) 2288 (c) 2294 (d) 2290
10. 10.Which of these numbers is divisible by 4? (a) 2733 (b) 2739 (c) 2732 (d) 2729
11. 11.What is the remainder when 119 is divided by 3? (a) 2 (b) 0 (c) 1
12. 12.What is the remainder when 85 is divided by 8? (a) 4 (b) 6 (c) 3 (d) 5
13. 13.What is the remainder when 304 is divided by 10? (a) 6 (b) 3 (c) 5 (d) 4
14. 14.Which of these numbers is divisible by 11? (a) 8753 (b) 8739 (c) 8759 (d) 8745
15. 15.Which of these numbers is divisible by 9? (a) 7814 (b) 7811 (c) 7799 (d) 7803
16. 16.Which of these numbers is divisible by 3? (a) 178 (b) 175 (c) 181 (d) 177
17. 17.What is the units digit of 9^20? (a) 1 (b) 9 (c) 2 (d) 0
18. 18.What is the units digit of 3^15? (a) 9 (b) 7 (c) 3 (d) 1
19. 19.What is the smallest number you must add to 410 to make it divisible by 12? (a) 9 (b) 11 (c) 2 (d) 10
20. 20.What is the units digit of 3^21? (a) 3 (b) 9 (c) 7 (d) 1
21. 21.What is the units digit of 3^19? (a) 1 (b) 3 (c) 7 (d) 9
22. 22.Which of these numbers is divisible by 3? (a) 1958 (b) 1956 (c) 1960 (d) 1954
23. 23.What is the units digit of 9^14? (a) 0 (b) 9 (c) 1 (d) 2
24. 24.What is the remainder when 313 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
25. 25.What is the units digit of 3^11? (a) 9 (b) 3 (c) 7 (d) 1
26. 26.What is the remainder when 264 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
27. 27.What is the remainder when 229 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
28. 28.Which of these numbers is divisible by 8? (a) 4368 (b) 4378 (c) 4363 (d) 4369
29. 29.What is the units digit of 8^36? (a) 6 (b) 8 (c) 2 (d) 4
30. 30.What is the remainder when 265 is divided by 11? (a) 10 (b) 2 (c) 0 (d) 1
31. 31.Which of these numbers is divisible by 3? (a) 608 (b) 611 (c) 606 (d) 605
32. 32.What is the remainder when 187 is divided by 3? (a) 0 (b) 1 (c) 2
33. 33.What is the units digit of 2^44? (a) 4 (b) 2 (c) 8 (d) 6
34. 34.What is the remainder when 166 is divided by 7? (a) 2 (b) 6 (c) 4 (d) 5
35. 35.What is the smallest number you must add to 215 to make it divisible by 7? (a) 1 (b) 2 (c) 3 (d) 5
36. 36.Which of these numbers is divisible by 3? (a) 1811 (b) 1813 (c) 1816 (d) 1812
37. 37.What is the remainder when 236 is divided by 9? (a) 3 (b) 2 (c) 1 (d) 7
38. 38.What is the remainder when 165 is divided by 12? (a) 9 (b) 10 (c) 3 (d) 8
39. 39.What is the units digit of 9^23? (a) 1 (b) 8 (c) 7 (d) 9
40. 40.What is the units digit of 8^13? (a) 2 (b) 6 (c) 4 (d) 8

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