Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 628 · 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 smallest number you must add to 99 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 1
2. 2.What is the smallest number you must add to 738 to make it divisible by 10? (a) 8 (b) 1 (c) 3 (d) 2
3. 3.Which of these numbers is divisible by 8? (a) 1957 (b) 1937 (c) 1944 (d) 1949
4. 4.Which of these numbers is divisible by 4? (a) 2560 (b) 2562 (c) 2567 (d) 2559
5. 5.What is the remainder when 380 is divided by 8? (a) 6 (b) 5 (c) 4 (d) 3
6. 6.What is the units digit of 8^25? (a) 4 (b) 8 (c) 6 (d) 2
7. 7.What is the units digit of 4^50? (a) 4 (b) 7 (c) 6 (d) 5
8. 8.What is the units digit of 7^55? (a) 7 (b) 9 (c) 3 (d) 1
9. 9.What is the remainder when 286 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
10. 10.What is the smallest number you must add to 332 to make it divisible by 6? (a) 2 (b) 4 (c) 3 (d) 5
11. 11.Which of these numbers is divisible by 3? (a) 683 (b) 684 (c) 688 (d) 686
12. 12.What is the remainder when 107 is divided by 3? (a) 0 (b) 2 (c) 1
13. 13.Which of these numbers is divisible by 9? (a) 5450 (b) 5445 (c) 5443 (d) 5458
14. 14.Which of these numbers is divisible by 4? (a) 3405 (b) 3404 (c) 3403 (d) 3409
15. 15.Which of these numbers is divisible by 8? (a) 6918 (b) 6911 (c) 6899 (d) 6904
16. 16.What is the smallest number you must add to 211 to make it divisible by 3? (a) 1 (b) 2
17. 17.Which of these numbers is divisible by 4? (a) 2806 (b) 2799 (c) 2800 (d) 2803
18. 18.What is the units digit of 7^54? (a) 9 (b) 7 (c) 3 (d) 1
19. 19.What is the smallest number you must add to 673 to make it divisible by 11? (a) 2 (b) 9 (c) 8 (d) 10
20. 20.Which of these numbers is divisible by 11? (a) 3253 (b) 3243 (c) 3225 (d) 3234
21. 21.Which of these numbers is divisible by 9? (a) 1364 (b) 1368 (c) 1372 (d) 1384
22. 22.What is the units digit of 4^55? (a) 6 (b) 5 (c) 4 (d) 3
23. 23.Which of these numbers is divisible by 4? (a) 3097 (b) 3100 (c) 3107 (d) 3102
24. 24.What is the units digit of 2^22? (a) 6 (b) 2 (c) 4 (d) 8
25. 25.What is the units digit of 7^36? (a) 7 (b) 3 (c) 9 (d) 1
26. 26.Which of these numbers is divisible by 8? (a) 1460 (b) 1448 (c) 1445 (d) 1451
27. 27.What is the units digit of 2^35? (a) 6 (b) 8 (c) 4 (d) 2
28. 28.What is the remainder when 197 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
29. 29.What is the units digit of 7^19? (a) 9 (b) 7 (c) 1 (d) 3
30. 30.What is the smallest number you must add to 214 to make it divisible by 6? (a) 2 (b) 4 (c) 3 (d) 1
31. 31.What is the remainder when 318 is divided by 10? (a) 9 (b) 8 (c) 7 (d) 2
32. 32.What is the smallest number you must add to 273 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 1
33. 33.What is the remainder when 309 is divided by 12? (a) 8 (b) 3 (c) 9 (d) 10
34. 34.What is the remainder when 639 is divided by 10? (a) 1 (b) 8 (c) 7 (d) 9
35. 35.What is the remainder when 753 is divided by 11? (a) 5 (b) 7 (c) 6 (d) 4
36. 36.Which of these numbers is divisible by 3? (a) 370 (b) 376 (c) 373 (d) 372
37. 37.What is the remainder when 107 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
38. 38.What is the remainder when 113 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
39. 39.Which of these numbers is divisible by 11? (a) 2874 (b) 2902 (c) 2883 (d) 2882
40. 40.What is the smallest number you must add to 668 to make it divisible by 11? (a) 2 (b) 8 (c) 4 (d) 3

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