Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 860 · 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 9^48? (a) 9 (b) 0 (c) 2 (d) 1
2. 2.Which of these numbers is divisible by 4? (a) 572 (b) 574 (c) 578 (d) 569
3. 3.Which of these numbers is divisible by 9? (a) 3135 (b) 3123 (c) 3126 (d) 3115
4. 4.Which of these numbers is divisible by 8? (a) 2976 (b) 2977 (c) 2971 (d) 2985
5. 5.What is the smallest number you must add to 741 to make it divisible by 10? (a) 1 (b) 9 (c) 7 (d) 8
6. 6.Which of these numbers is divisible by 4? (a) 691 (b) 693 (c) 699 (d) 692
7. 7.Which of these numbers is divisible by 4? (a) 2119 (b) 2115 (c) 2112 (d) 2109
8. 8.What is the smallest number you must add to 466 to make it divisible by 7? (a) 4 (b) 5 (c) 2 (d) 3
9. 9.Which of these numbers is divisible by 4? (a) 1104 (b) 1101 (c) 1105 (d) 1110
10. 10.What is the units digit of 2^32? (a) 2 (b) 4 (c) 6 (d) 8
11. 11.What is the smallest number you must add to 736 to make it divisible by 11? (a) 10 (b) 3 (c) 1 (d) 2
12. 12.What is the smallest number you must add to 590 to make it divisible by 9? (a) 5 (b) 3 (c) 6 (d) 4
13. 13.Which of these numbers is divisible by 9? (a) 4299 (b) 4307 (c) 4319 (d) 4302
14. 14.Which of these numbers is divisible by 9? (a) 695 (b) 704 (c) 686 (d) 693
15. 15.What is the units digit of 9^19? (a) 7 (b) 9 (c) 8 (d) 1
16. 16.Which of these numbers is divisible by 9? (a) 1979 (b) 1969 (c) 1971 (d) 1988
17. 17.What is the units digit of 8^56? (a) 4 (b) 2 (c) 8 (d) 6
18. 18.Which of these numbers is divisible by 11? (a) 7159 (b) 7165 (c) 7179 (d) 7161
19. 19.Which of these numbers is divisible by 11? (a) 9042 (b) 9036 (c) 9052 (d) 9063
20. 20.What is the remainder when 627 is divided by 8? (a) 5 (b) 3 (c) 2 (d) 4
21. 21.Which of these numbers is divisible by 11? (a) 4776 (b) 4753 (c) 4771 (d) 4763
22. 22.What is the remainder when 71 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
23. 23.What is the remainder when 721 is divided by 10? (a) 0 (b) 9 (c) 1 (d) 2
24. 24.What is the units digit of 3^13? (a) 9 (b) 1 (c) 3 (d) 7
25. 25.Which of these numbers is divisible by 4? (a) 2706 (b) 2698 (c) 2700 (d) 2701
26. 26.What is the smallest number you must add to 259 to make it divisible by 8? (a) 5 (b) 4 (c) 6 (d) 3
27. 27.What is the remainder when 71 is divided by 3? (a) 2 (b) 1 (c) 0
28. 28.What is the units digit of 9^39? (a) 1 (b) 7 (c) 8 (d) 9
29. 29.What is the remainder when 676 is divided by 11? (a) 6 (b) 7 (c) 5 (d) 4
30. 30.What is the smallest number you must add to 89 to make it divisible by 3? (a) 2 (b) 1
31. 31.What is the units digit of 7^39? (a) 7 (b) 9 (c) 3 (d) 1
32. 32.What is the units digit of 2^38? (a) 4 (b) 6 (c) 8 (d) 2
33. 33.What is the units digit of 7^46? (a) 7 (b) 9 (c) 3 (d) 1
34. 34.What is the smallest number you must add to 265 to make it divisible by 8? (a) 7 (b) 1 (c) 5 (d) 6
35. 35.What is the remainder when 362 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
36. 36.What is the smallest number you must add to 261 to make it divisible by 8? (a) 2 (b) 4 (c) 3 (d) 5
37. 37.What is the smallest number you must add to 219 to make it divisible by 4? (a) 2 (b) 3 (c) 1
38. 38.What is the units digit of 2^29? (a) 4 (b) 2 (c) 8 (d) 6
39. 39.What is the smallest number you must add to 391 to make it divisible by 9? (a) 7 (b) 4 (c) 5 (d) 6
40. 40.What is the units digit of 9^20? (a) 0 (b) 1 (c) 2 (d) 9

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