Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 471 · 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 848 is divided by 12? (a) 8 (b) 7 (c) 4 (d) 9
2. 2.What is the units digit of 4^58? (a) 7 (b) 6 (c) 4 (d) 5
3. 3.What is the remainder when 54 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
4. 4.What is the units digit of 3^29? (a) 1 (b) 9 (c) 7 (d) 3
5. 5.What is the units digit of 8^41? (a) 6 (b) 4 (c) 2 (d) 8
6. 6.Which of these numbers is divisible by 8? (a) 6110 (b) 6115 (c) 6100 (d) 6104
7. 7.What is the units digit of 4^19? (a) 5 (b) 4 (c) 3 (d) 6
8. 8.What is the smallest number you must add to 637 to make it divisible by 8? (a) 2 (b) 3 (c) 5 (d) 4
9. 9.What is the units digit of 2^28? (a) 2 (b) 8 (c) 4 (d) 6
10. 10.What is the smallest number you must add to 567 to make it divisible by 8? (a) 7 (b) 2 (c) 3 (d) 1
11. 11.Which of these numbers is divisible by 9? (a) 4879 (b) 4890 (c) 4897 (d) 4887
12. 12.What is the remainder when 269 is divided by 11? (a) 4 (b) 6 (c) 7 (d) 5
13. 13.What is the units digit of 2^38? (a) 2 (b) 6 (c) 4 (d) 8
14. 14.Which of these numbers is divisible by 4? (a) 2545 (b) 2554 (c) 2549 (d) 2548
15. 15.What is the remainder when 43 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
16. 16.What is the remainder when 652 is divided by 11? (a) 2 (b) 3 (c) 4 (d) 8
17. 17.What is the smallest number you must add to 370 to make it divisible by 7? (a) 3 (b) 6 (c) 1 (d) 2
18. 18.What is the smallest number you must add to 47 to make it divisible by 4? (a) 3 (b) 2 (c) 1
19. 19.What is the units digit of 9^50? (a) 2 (b) 0 (c) 1 (d) 9
20. 20.What is the remainder when 230 is divided by 3? (a) 0 (b) 1 (c) 2
21. 21.What is the units digit of 9^32? (a) 9 (b) 0 (c) 1 (d) 2
22. 22.What is the remainder when 89 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
23. 23.What is the remainder when 547 is divided by 8? (a) 3 (b) 2 (c) 4 (d) 5
24. 24.What is the units digit of 4^41? (a) 3 (b) 5 (c) 6 (d) 4
25. 25.What is the units digit of 3^18? (a) 9 (b) 7 (c) 3 (d) 1
26. 26.What is the units digit of 2^25? (a) 4 (b) 8 (c) 2 (d) 6
27. 27.What is the remainder when 294 is divided by 11? (a) 7 (b) 8 (c) 3 (d) 9
28. 28.Which of these numbers is divisible by 4? (a) 2318 (b) 2322 (c) 2320 (d) 2326
29. 29.What is the smallest number you must add to 200 to make it divisible by 3? (a) 1 (b) 2
30. 30.What is the units digit of 8^10? (a) 8 (b) 6 (c) 2 (d) 4
31. 31.What is the smallest number you must add to 85 to make it divisible by 3? (a) 2 (b) 1
32. 32.Which of these numbers is divisible by 11? (a) 895 (b) 872 (c) 880 (d) 890
33. 33.Which of these numbers is divisible by 3? (a) 1870 (b) 1872 (c) 1873 (d) 1876
34. 34.What is the remainder when 256 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
35. 35.Which of these numbers is divisible by 8? (a) 7136 (b) 7139 (c) 7147 (d) 7134
36. 36.Which of these numbers is divisible by 11? (a) 9234 (b) 9229 (c) 9245 (d) 9224
37. 37.Which of these numbers is divisible by 4? (a) 1441 (b) 1451 (c) 1444 (d) 1445
38. 38.What is the remainder when 377 is divided by 7? (a) 1 (b) 4 (c) 6 (d) 5
39. 39.What is the remainder when 155 is divided by 10? (a) 6 (b) 7 (c) 4 (d) 5
40. 40.What is the smallest number you must add to 424 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 1

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