Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 338 · 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.Which of these numbers is divisible by 9? (a) 4435 (b) 4441 (c) 4437 (d) 4450
2. 2.Which of these numbers is divisible by 9? (a) 7137 (b) 7152 (c) 7142 (d) 7131
3. 3.What is the units digit of 3^45? (a) 1 (b) 9 (c) 7 (d) 3
4. 4.Which of these numbers is divisible by 8? (a) 3774 (b) 3761 (c) 3760 (d) 3759
5. 5.What is the smallest number you must add to 237 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
6. 6.What is the remainder when 124 is divided by 3? (a) 2 (b) 1 (c) 0
7. 7.What is the remainder when 542 is divided by 7? (a) 4 (b) 5 (c) 2 (d) 3
8. 8.What is the remainder when 665 is divided by 10? (a) 7 (b) 6 (c) 5 (d) 4
9. 9.What is the units digit of 7^46? (a) 7 (b) 1 (c) 3 (d) 9
10. 10.Which of these numbers is divisible by 9? (a) 3900 (b) 3907 (c) 3916 (d) 3906
11. 11.Which of these numbers is divisible by 3? (a) 2072 (b) 2068 (c) 2067 (d) 2065
12. 12.What is the remainder when 239 is divided by 7? (a) 0 (b) 2 (c) 1 (d) 6
13. 13.Which of these numbers is divisible by 4? (a) 2135 (b) 2139 (c) 2136 (d) 2141
14. 14.What is the remainder when 100 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
15. 15.What is the remainder when 306 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
16. 16.What is the remainder when 770 is divided by 12? (a) 3 (b) 1 (c) 2 (d) 10
17. 17.What is the units digit of 8^13? (a) 2 (b) 6 (c) 4 (d) 8
18. 18.Which of these numbers is divisible by 6? (a) 5164 (b) 5157 (c) 5170 (d) 5160
19. 19.What is the units digit of 2^13? (a) 8 (b) 4 (c) 2 (d) 6
20. 20.What is the smallest number you must add to 448 to make it divisible by 9? (a) 3 (b) 7 (c) 1 (d) 2
21. 21.What is the remainder when 661 is divided by 11? (a) 10 (b) 1 (c) 2 (d) 0
22. 22.What is the smallest number you must add to 538 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 6
23. 23.What is the remainder when 226 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
24. 24.Which of these numbers is divisible by 4? (a) 3517 (b) 3515 (c) 3522 (d) 3516
25. 25.What is the smallest number you must add to 103 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
26. 26.What is the remainder when 116 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 1
27. 27.What is the smallest number you must add to 259 to make it divisible by 11? (a) 6 (b) 4 (c) 5 (d) 7
28. 28.What is the remainder when 316 is divided by 9? (a) 8 (b) 1 (c) 0 (d) 2
29. 29.What is the smallest number you must add to 79 to make it divisible by 4? (a) 3 (b) 2 (c) 1
30. 30.What is the smallest number you must add to 238 to make it divisible by 3? (a) 1 (b) 2
31. 31.What is the remainder when 480 is divided by 9? (a) 3 (b) 2 (c) 6 (d) 4
32. 32.What is the remainder when 167 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
33. 33.What is the units digit of 7^13? (a) 1 (b) 7 (c) 9 (d) 3
34. 34.What is the remainder when 416 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
35. 35.Which of these numbers is divisible by 4? (a) 2571 (b) 2573 (c) 2568 (d) 2566
36. 36.What is the smallest number you must add to 534 to make it divisible by 9? (a) 7 (b) 5 (c) 3 (d) 6
37. 37.What is the smallest number you must add to 476 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
38. 38.What is the units digit of 8^56? (a) 2 (b) 4 (c) 8 (d) 6
39. 39.What is the units digit of 3^59? (a) 1 (b) 9 (c) 7 (d) 3
40. 40.What is the smallest number you must add to 118 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2

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