Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 144 · 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 362 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
2. 2.What is the remainder when 377 is divided by 8? (a) 7 (b) 2 (c) 0 (d) 1
3. 3.What is the units digit of 4^23? (a) 5 (b) 3 (c) 6 (d) 4
4. 4.What is the remainder when 327 is divided by 12? (a) 9 (b) 2 (c) 4 (d) 3
5. 5.What is the remainder when 715 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 7
6. 6.Which of these numbers is divisible by 3? (a) 2478 (b) 2480 (c) 2483 (d) 2477
7. 7.Which of these numbers is divisible by 11? (a) 3421 (b) 3412 (c) 3439 (d) 3424
8. 8.What is the smallest number you must add to 459 to make it divisible by 7? (a) 3 (b) 5 (c) 4 (d) 2
9. 9.What is the smallest number you must add to 391 to make it divisible by 11? (a) 7 (b) 6 (c) 4 (d) 5
10. 10.What is the units digit of 4^15? (a) 5 (b) 4 (c) 3 (d) 6
11. 11.What is the smallest number you must add to 119 to make it divisible by 4? (a) 2 (b) 1 (c) 3
12. 12.What is the smallest number you must add to 491 to make it divisible by 7? (a) 5 (b) 4 (c) 1 (d) 6
13. 13.What is the smallest number you must add to 98 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 4
14. 14.What is the remainder when 102 is divided by 9? (a) 6 (b) 2 (c) 4 (d) 3
15. 15.What is the remainder when 180 is divided by 8? (a) 4 (b) 6 (c) 5 (d) 3
16. 16.What is the smallest number you must add to 288 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
17. 17.What is the remainder when 122 is divided by 6? (a) 1 (b) 3 (c) 2 (d) 4
18. 18.Which of these numbers is divisible by 3? (a) 1429 (b) 1426 (c) 1432 (d) 1428
19. 19.What is the smallest number you must add to 414 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 6
20. 20.What is the units digit of 3^35? (a) 3 (b) 1 (c) 7 (d) 9
21. 21.What is the remainder when 497 is divided by 11? (a) 9 (b) 2 (c) 1 (d) 3
22. 22.What is the remainder when 490 is divided by 9? (a) 4 (b) 3 (c) 5 (d) 6
23. 23.What is the remainder when 353 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
24. 24.Which of these numbers is divisible by 11? (a) 9205 (b) 9195 (c) 9185 (d) 9176
25. 25.Which of these numbers is divisible by 9? (a) 6656 (b) 6660 (c) 6670 (d) 6668
26. 26.What is the remainder when 548 is divided by 9? (a) 8 (b) 6 (c) 1 (d) 7
27. 27.Which of these numbers is divisible by 6? (a) 4392 (b) 4390 (c) 4396 (d) 4401
28. 28.What is the smallest number you must add to 299 to make it divisible by 4? (a) 1 (b) 3 (c) 2
29. 29.Which of these numbers is divisible by 3? (a) 751 (b) 750 (c) 748 (d) 754
30. 30.Which of these numbers is divisible by 4? (a) 874 (b) 865 (c) 869 (d) 868
31. 31.What is the remainder when 208 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 2
32. 32.What is the units digit of 4^22? (a) 4 (b) 6 (c) 7 (d) 5
33. 33.Which of these numbers is divisible by 6? (a) 3064 (b) 3067 (c) 3073 (d) 3066
34. 34.What is the remainder when 149 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 1
35. 35.What is the remainder when 410 is divided by 8? (a) 1 (b) 6 (c) 2 (d) 3
36. 36.What is the units digit of 3^16? (a) 3 (b) 1 (c) 7 (d) 9
37. 37.Which of these numbers is divisible by 3? (a) 1493 (b) 1496 (c) 1498 (d) 1494
38. 38.Which of these numbers is divisible by 3? (a) 427 (b) 430 (c) 429 (d) 434
39. 39.Which of these numbers is divisible by 8? (a) 2800 (b) 2815 (c) 2798 (d) 2807
40. 40.What is the smallest number you must add to 163 to make it divisible by 6? (a) 1 (b) 3 (c) 4 (d) 5

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