Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 772 · 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 448 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
2. 2.Which of these numbers is divisible by 8? (a) 987 (b) 982 (c) 993 (d) 984
3. 3.What is the remainder when 457 is divided by 6? (a) 0 (b) 5 (c) 1 (d) 2
4. 4.Which of these numbers is divisible by 4? (a) 3120 (b) 3121 (c) 3117 (d) 3126
5. 5.What is the smallest number you must add to 181 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
6. 6.What is the units digit of 4^38? (a) 5 (b) 7 (c) 6 (d) 4
7. 7.What is the smallest number you must add to 185 to make it divisible by 4? (a) 3 (b) 2 (c) 1
8. 8.What is the smallest number you must add to 242 to make it divisible by 3? (a) 1 (b) 2
9. 9.What is the remainder when 747 is divided by 12? (a) 3 (b) 9 (c) 2 (d) 4
10. 10.What is the remainder when 271 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
11. 11.What is the remainder when 149 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 1
12. 12.What is the smallest number you must add to 641 to make it divisible by 10? (a) 9 (b) 8 (c) 7 (d) 1
13. 13.What is the units digit of 2^14? (a) 6 (b) 2 (c) 4 (d) 8
14. 14.What is the smallest number you must add to 219 to make it divisible by 4? (a) 1 (b) 2 (c) 3
15. 15.Which of these numbers is divisible by 8? (a) 586 (b) 595 (c) 605 (d) 592
16. 16.Which of these numbers is divisible by 4? (a) 1009 (b) 1014 (c) 1012 (d) 1019
17. 17.What is the smallest number you must add to 233 to make it divisible by 4? (a) 2 (b) 3 (c) 1
18. 18.What is the units digit of 9^27? (a) 9 (b) 7 (c) 1 (d) 8
19. 19.Which of these numbers is divisible by 4? (a) 1973 (b) 1968 (c) 1966 (d) 1969
20. 20.What is the remainder when 91 is divided by 9? (a) 0 (b) 2 (c) 1 (d) 8
21. 21.What is the smallest number you must add to 241 to make it divisible by 4? (a) 2 (b) 1 (c) 3
22. 22.Which of these numbers is divisible by 11? (a) 3837 (b) 3839 (c) 3848 (d) 3855
23. 23.What is the remainder when 71 is divided by 3? (a) 2 (b) 1 (c) 0
24. 24.What is the smallest number you must add to 175 to make it divisible by 3? (a) 1 (b) 2
25. 25.What is the remainder when 278 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 2
26. 26.What is the smallest number you must add to 133 to make it divisible by 9? (a) 3 (b) 2 (c) 1 (d) 7
27. 27.What is the remainder when 307 is divided by 11? (a) 1 (b) 9 (c) 10 (d) 8
28. 28.Which of these numbers is divisible by 9? (a) 510 (b) 495 (c) 494 (d) 502
29. 29.Which of these numbers is divisible by 8? (a) 5615 (b) 5601 (c) 5621 (d) 5608
30. 30.What is the smallest number you must add to 628 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
31. 31.Which of these numbers is divisible by 9? (a) 6382 (b) 6404 (c) 6392 (d) 6390
32. 32.What is the units digit of 4^57? (a) 4 (b) 5 (c) 6 (d) 3
33. 33.What is the remainder when 434 is divided by 9? (a) 7 (b) 1 (c) 3 (d) 2
34. 34.What is the units digit of 8^26? (a) 2 (b) 8 (c) 6 (d) 4
35. 35.Which of these numbers is divisible by 3? (a) 2603 (b) 2606 (c) 2609 (d) 2604
36. 36.Which of these numbers is divisible by 6? (a) 3674 (b) 3680 (c) 3688 (d) 3678
37. 37.What is the units digit of 7^11? (a) 1 (b) 7 (c) 3 (d) 9
38. 38.What is the smallest number you must add to 174 to make it divisible by 9? (a) 6 (b) 7 (c) 3 (d) 5
39. 39.What is the units digit of 9^14? (a) 1 (b) 2 (c) 9 (d) 0
40. 40.Which of these numbers is divisible by 3? (a) 295 (b) 290 (c) 293 (d) 291

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