Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 864 · 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.Which of these numbers is divisible by 3? (a) 2452 (b) 2455 (c) 2454 (d) 2459
2. 2.Which of these numbers is divisible by 8? (a) 622 (b) 612 (c) 608 (d) 607
3. 3.Which of these numbers is divisible by 8? (a) 3689 (b) 3682 (c) 3680 (d) 3674
4. 4.Which of these numbers is divisible by 3? (a) 1693 (b) 1692 (c) 1697 (d) 1690
5. 5.What is the smallest number you must add to 673 to make it divisible by 10? (a) 6 (b) 8 (c) 3 (d) 7
6. 6.Which of these numbers is divisible by 3? (a) 551 (b) 557 (c) 552 (d) 553
7. 7.What is the units digit of 8^44? (a) 2 (b) 6 (c) 8 (d) 4
8. 8.Which of these numbers is divisible by 4? (a) 3217 (b) 3223 (c) 3213 (d) 3216
9. 9.What is the units digit of 3^31? (a) 3 (b) 7 (c) 9 (d) 1
10. 10.What is the units digit of 8^12? (a) 4 (b) 8 (c) 2 (d) 6
11. 11.What is the remainder when 141 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
12. 12.What is the units digit of 4^20? (a) 4 (b) 5 (c) 6 (d) 7
13. 13.What is the remainder when 275 is divided by 9? (a) 6 (b) 7 (c) 5 (d) 4
14. 14.What is the remainder when 385 is divided by 6? (a) 5 (b) 1 (c) 0 (d) 2
15. 15.What is the units digit of 2^28? (a) 8 (b) 6 (c) 2 (d) 4
16. 16.What is the smallest number you must add to 345 to make it divisible by 12? (a) 3 (b) 9 (c) 2 (d) 4
17. 17.Which of these numbers is divisible by 6? (a) 3969 (b) 3962 (c) 3960 (d) 3959
18. 18.What is the remainder when 209 is divided by 9? (a) 7 (b) 2 (c) 1 (d) 3
19. 19.What is the units digit of 2^33? (a) 4 (b) 2 (c) 8 (d) 6
20. 20.What is the remainder when 44 is divided by 3? (a) 1 (b) 2 (c) 0
21. 21.What is the remainder when 261 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
22. 22.What is the remainder when 88 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 2
23. 23.What is the units digit of 3^13? (a) 3 (b) 9 (c) 7 (d) 1
24. 24.What is the remainder when 171 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
25. 25.What is the remainder when 172 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
26. 26.Which of these numbers is divisible by 3? (a) 1933 (b) 1940 (c) 1936 (d) 1935
27. 27.What is the remainder when 337 is divided by 9? (a) 6 (b) 3 (c) 5 (d) 4
28. 28.What is the smallest number you must add to 323 to make it divisible by 4? (a) 1 (b) 2 (c) 3
29. 29.What is the units digit of 4^26? (a) 4 (b) 7 (c) 6 (d) 5
30. 30.What is the remainder when 567 is divided by 8? (a) 5 (b) 1 (c) 6 (d) 7
31. 31.What is the smallest number you must add to 441 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
32. 32.Which of these numbers is divisible by 9? (a) 3006 (b) 3003 (c) 3023 (d) 3011
33. 33.What is the remainder when 781 is divided by 10? (a) 1 (b) 9 (c) 2 (d) 0
34. 34.Which of these numbers is divisible by 6? (a) 653 (b) 642 (c) 637 (d) 646
35. 35.What is the remainder when 161 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
36. 36.What is the remainder when 385 is divided by 9? (a) 6 (b) 8 (c) 7 (d) 2
37. 37.Which of these numbers is divisible by 6? (a) 3611 (b) 3604 (c) 3614 (d) 3606
38. 38.What is the remainder when 723 is divided by 11? (a) 9 (b) 3 (c) 8 (d) 7
39. 39.Which of these numbers is divisible by 8? (a) 6837 (b) 6840 (c) 6849 (d) 6846
40. 40.Which of these numbers is divisible by 3? (a) 188 (b) 186 (c) 185 (d) 191

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