Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 496 · 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 298 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
2. 2.Which of these numbers is divisible by 11? (a) 611 (b) 585 (c) 594 (d) 598
3. 3.Which of these numbers is divisible by 4? (a) 710 (b) 701 (c) 704 (d) 705
4. 4.What is the smallest number you must add to 485 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 6
5. 5.What is the remainder when 262 is divided by 10? (a) 8 (b) 3 (c) 1 (d) 2
6. 6.Which of these numbers is divisible by 8? (a) 5263 (b) 5248 (c) 5243 (d) 5255
7. 7.Which of these numbers is divisible by 3? (a) 2323 (b) 2329 (c) 2325 (d) 2327
8. 8.What is the remainder when 430 is divided by 6? (a) 5 (b) 3 (c) 2 (d) 4
9. 9.What is the remainder when 793 is divided by 12? (a) 11 (b) 2 (c) 0 (d) 1
10. 10.What is the units digit of 8^13? (a) 2 (b) 8 (c) 6 (d) 4
11. 11.Which of these numbers is divisible by 4? (a) 2956 (b) 2954 (c) 2961 (d) 2957
12. 12.What is the remainder when 145 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
13. 13.Which of these numbers is divisible by 3? (a) 1453 (b) 1451 (c) 1449 (d) 1447
14. 14.What is the remainder when 128 is divided by 6? (a) 3 (b) 1 (c) 4 (d) 2
15. 15.What is the remainder when 494 is divided by 9? (a) 8 (b) 6 (c) 7 (d) 1
16. 16.What is the remainder when 612 is divided by 8? (a) 5 (b) 3 (c) 4 (d) 6
17. 17.What is the remainder when 232 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
18. 18.What is the smallest number you must add to 52 to make it divisible by 3? (a) 1 (b) 2
19. 19.Which of these numbers is divisible by 11? (a) 2185 (b) 2167 (c) 2175 (d) 2158
20. 20.What is the remainder when 83 is divided by 3? (a) 1 (b) 2 (c) 0
21. 21.Which of these numbers is divisible by 6? (a) 4444 (b) 4431 (c) 4434 (d) 4435
22. 22.What is the remainder when 180 is divided by 11? (a) 7 (b) 4 (c) 3 (d) 5
23. 23.What is the smallest number you must add to 596 to make it divisible by 10? (a) 3 (b) 4 (c) 5 (d) 6
24. 24.What is the smallest number you must add to 68 to make it divisible by 3? (a) 2 (b) 1
25. 25.What is the units digit of 2^56? (a) 8 (b) 4 (c) 2 (d) 6
26. 26.What is the smallest number you must add to 554 to make it divisible by 9? (a) 5 (b) 6 (c) 3 (d) 4
27. 27.What is the smallest number you must add to 192 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
28. 28.What is the units digit of 8^20? (a) 4 (b) 2 (c) 8 (d) 6
29. 29.Which of these numbers is divisible by 9? (a) 2417 (b) 2407 (c) 2427 (d) 2412
30. 30.What is the smallest number you must add to 188 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
31. 31.What is the remainder when 267 is divided by 10? (a) 7 (b) 8 (c) 3 (d) 6
32. 32.What is the units digit of 7^28? (a) 7 (b) 3 (c) 1 (d) 9
33. 33.What is the smallest number you must add to 314 to make it divisible by 9? (a) 2 (b) 8 (c) 3 (d) 1
34. 34.What is the units digit of 2^38? (a) 2 (b) 6 (c) 4 (d) 8
35. 35.What is the units digit of 7^14? (a) 9 (b) 3 (c) 1 (d) 7
36. 36.What is the smallest number you must add to 161 to make it divisible by 8? (a) 7 (b) 6 (c) 5 (d) 1
37. 37.What is the remainder when 241 is divided by 7? (a) 4 (b) 2 (c) 3 (d) 5
38. 38.What is the smallest number you must add to 400 to make it divisible by 7? (a) 1 (b) 5 (c) 4 (d) 6
39. 39.What is the smallest number you must add to 102 to make it divisible by 8? (a) 2 (b) 6 (c) 1 (d) 3
40. 40.What is the remainder when 187 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3

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