Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 81 · 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 99 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
2. 2.What is the smallest number you must add to 172 to make it divisible by 3? (a) 1 (b) 2
3. 3.What is the remainder when 205 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
4. 4.What is the smallest number you must add to 324 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
5. 5.Which of these numbers is divisible by 4? (a) 3579 (b) 3573 (c) 3572 (d) 3571
6. 6.What is the remainder when 310 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
7. 7.Which of these numbers is divisible by 4? (a) 2885 (b) 2881 (c) 2878 (d) 2880
8. 8.Which of these numbers is divisible by 8? (a) 5590 (b) 5605 (c) 5592 (d) 5596
9. 9.Which of these numbers is divisible by 4? (a) 2565 (b) 2563 (c) 2560 (d) 2558
10. 10.Which of these numbers is divisible by 11? (a) 6059 (b) 6039 (c) 6036 (d) 6046
11. 11.What is the smallest number you must add to 409 to make it divisible by 12? (a) 9 (b) 1 (c) 11 (d) 10
12. 12.Which of these numbers is divisible by 3? (a) 1288 (b) 1291 (c) 1287 (d) 1285
13. 13.What is the units digit of 8^27? (a) 8 (b) 4 (c) 6 (d) 2
14. 14.Which of these numbers is divisible by 8? (a) 4005 (b) 3992 (c) 3988 (d) 3999
15. 15.What is the smallest number you must add to 73 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
16. 16.Which of these numbers is divisible by 9? (a) 7829 (b) 7821 (c) 7815 (d) 7832
17. 17.Which of these numbers is divisible by 4? (a) 3320 (b) 3319 (c) 3321 (d) 3325
18. 18.What is the smallest number you must add to 197 to make it divisible by 3? (a) 1 (b) 2
19. 19.What is the units digit of 4^20? (a) 4 (b) 5 (c) 6 (d) 7
20. 20.Which of these numbers is divisible by 4? (a) 2326 (b) 2324 (c) 2329 (d) 2321
21. 21.Which of these numbers is divisible by 4? (a) 3263 (b) 3253 (c) 3258 (d) 3256
22. 22.Which of these numbers is divisible by 3? (a) 2299 (b) 2294 (c) 2295 (d) 2296
23. 23.What is the units digit of 8^51? (a) 8 (b) 2 (c) 4 (d) 6
24. 24.What is the units digit of 3^11? (a) 7 (b) 9 (c) 1 (d) 3
25. 25.What is the smallest number you must add to 693 to make it divisible by 12? (a) 2 (b) 9 (c) 3 (d) 4
26. 26.What is the smallest number you must add to 127 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 4
27. 27.What is the units digit of 4^38? (a) 7 (b) 4 (c) 6 (d) 5
28. 28.What is the smallest number you must add to 241 to make it divisible by 3? (a) 2 (b) 1
29. 29.Which of these numbers is divisible by 4? (a) 2837 (b) 2843 (c) 2840 (d) 2846
30. 30.What is the remainder when 679 is divided by 11? (a) 8 (b) 9 (c) 3 (d) 7
31. 31.What is the remainder when 510 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 1
32. 32.What is the units digit of 3^38? (a) 9 (b) 1 (c) 3 (d) 7
33. 33.What is the smallest number you must add to 186 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
34. 34.What is the smallest number you must add to 143 to make it divisible by 4? (a) 3 (b) 1 (c) 2
35. 35.Which of these numbers is divisible by 11? (a) 9066 (b) 9050 (c) 9061 (d) 9053
36. 36.Which of these numbers is divisible by 11? (a) 7419 (b) 7403 (c) 7400 (d) 7411
37. 37.What is the units digit of 2^32? (a) 2 (b) 4 (c) 6 (d) 8
38. 38.What is the units digit of 8^21? (a) 4 (b) 2 (c) 6 (d) 8
39. 39.What is the smallest number you must add to 498 to make it divisible by 8? (a) 5 (b) 6 (c) 7 (d) 2
40. 40.Which of these numbers is divisible by 4? (a) 570 (b) 565 (c) 564 (d) 563

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