Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 280 · 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 258 to make it divisible by 4? (a) 3 (b) 2 (c) 1
2. 2.What is the remainder when 61 is divided by 3? (a) 1 (b) 0 (c) 2
3. 3.What is the remainder when 388 is divided by 6? (a) 3 (b) 2 (c) 5 (d) 4
4. 4.What is the smallest number you must add to 114 to make it divisible by 7? (a) 5 (b) 6 (c) 2 (d) 4
5. 5.Which of these numbers is divisible by 9? (a) 2090 (b) 2101 (c) 2088 (d) 2085
6. 6.What is the remainder when 69 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
7. 7.Which of these numbers is divisible by 9? (a) 2663 (b) 2647 (c) 2639 (d) 2646
8. 8.What is the remainder when 422 is divided by 7? (a) 3 (b) 1 (c) 2 (d) 5
9. 9.What is the units digit of 7^14? (a) 7 (b) 1 (c) 3 (d) 9
10. 10.Which of these numbers is divisible by 3? (a) 1865 (b) 1866 (c) 1871 (d) 1867
11. 11.What is the remainder when 207 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
12. 12.What is the smallest number you must add to 89 to make it divisible by 4? (a) 2 (b) 3 (c) 1
13. 13.What is the units digit of 9^46? (a) 2 (b) 1 (c) 9 (d) 0
14. 14.What is the remainder when 534 is divided by 12? (a) 7 (b) 8 (c) 6 (d) 5
15. 15.Which of these numbers is divisible by 4? (a) 370 (b) 368 (c) 367 (d) 375
16. 16.Which of these numbers is divisible by 8? (a) 1367 (b) 1352 (c) 1346 (d) 1353
17. 17.What is the units digit of 7^28? (a) 3 (b) 9 (c) 7 (d) 1
18. 18.What is the smallest number you must add to 533 to make it divisible by 10? (a) 3 (b) 8 (c) 6 (d) 7
19. 19.What is the remainder when 528 is divided by 7? (a) 3 (b) 5 (c) 4 (d) 2
20. 20.What is the remainder when 412 is divided by 7? (a) 5 (b) 6 (c) 1 (d) 4
21. 21.What is the remainder when 239 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
22. 22.What is the units digit of 7^13? (a) 3 (b) 9 (c) 1 (d) 7
23. 23.What is the remainder when 41 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
24. 24.Which of these numbers is divisible by 6? (a) 2434 (b) 2437 (c) 2430 (d) 2428
25. 25.What is the remainder when 122 is divided by 11? (a) 2 (b) 0 (c) 10 (d) 1
26. 26.What is the remainder when 104 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 1
27. 27.Which of these numbers is divisible by 9? (a) 2783 (b) 2780 (c) 2767 (d) 2772
28. 28.Which of these numbers is divisible by 11? (a) 563 (b) 543 (c) 550 (d) 551
29. 29.What is the smallest number you must add to 321 to make it divisible by 9? (a) 2 (b) 4 (c) 3 (d) 6
30. 30.What is the remainder when 229 is divided by 11? (a) 9 (b) 10 (c) 2 (d) 8
31. 31.What is the remainder when 225 is divided by 11? (a) 5 (b) 7 (c) 6 (d) 4
32. 32.What is the smallest number you must add to 105 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
33. 33.What is the smallest number you must add to 346 to make it divisible by 11? (a) 5 (b) 6 (c) 8 (d) 7
34. 34.What is the remainder when 83 is divided by 6? (a) 3 (b) 5 (c) 1 (d) 4
35. 35.What is the units digit of 2^34? (a) 8 (b) 2 (c) 6 (d) 4
36. 36.What is the units digit of 4^55? (a) 5 (b) 3 (c) 4 (d) 6
37. 37.What is the remainder when 362 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 2
38. 38.What is the units digit of 2^19? (a) 6 (b) 8 (c) 4 (d) 2
39. 39.Which of these numbers is divisible by 11? (a) 1152 (b) 1137 (c) 1156 (d) 1144
40. 40.What is the remainder when 837 is divided by 12? (a) 8 (b) 3 (c) 9 (d) 10

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