Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 755 · 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 260 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 1
2. 2.What is the smallest number you must add to 399 to make it divisible by 9? (a) 5 (b) 7 (c) 3 (d) 6
3. 3.What is the smallest number you must add to 316 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
4. 4.What is the units digit of 3^38? (a) 3 (b) 9 (c) 7 (d) 1
5. 5.What is the smallest number you must add to 54 to make it divisible by 4? (a) 3 (b) 2 (c) 1
6. 6.What is the remainder when 628 is divided by 11? (a) 1 (b) 10 (c) 2 (d) 0
7. 7.What is the remainder when 760 is divided by 11? (a) 1 (b) 10 (c) 2 (d) 0
8. 8.What is the units digit of 8^18? (a) 2 (b) 6 (c) 4 (d) 8
9. 9.Which of these numbers is divisible by 6? (a) 3259 (b) 3269 (c) 3258 (d) 3255
10. 10.What is the remainder when 388 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 2
11. 11.What is the smallest number you must add to 629 to make it divisible by 9? (a) 3 (b) 1 (c) 8 (d) 2
12. 12.Which of these numbers is divisible by 11? (a) 6843 (b) 6836 (c) 6827 (d) 6831
13. 13.What is the remainder when 555 is divided by 11? (a) 7 (b) 5 (c) 6 (d) 4
14. 14.What is the units digit of 7^28? (a) 9 (b) 1 (c) 7 (d) 3
15. 15.What is the smallest number you must add to 133 to make it divisible by 3? (a) 2 (b) 1
16. 16.What is the smallest number you must add to 550 to make it divisible by 8? (a) 3 (b) 6 (c) 2 (d) 1
17. 17.What is the remainder when 308 is divided by 9? (a) 3 (b) 7 (c) 2 (d) 1
18. 18.Which of these numbers is divisible by 4? (a) 2368 (b) 2370 (c) 2367 (d) 2375
19. 19.Which of these numbers is divisible by 11? (a) 3392 (b) 3408 (c) 3414 (d) 3399
20. 20.What is the units digit of 3^34? (a) 1 (b) 7 (c) 3 (d) 9
21. 21.What is the units digit of 9^21? (a) 9 (b) 7 (c) 1 (d) 8
22. 22.What is the remainder when 357 is divided by 9? (a) 7 (b) 5 (c) 3 (d) 6
23. 23.What is the smallest number you must add to 285 to make it divisible by 7? (a) 3 (b) 5 (c) 2 (d) 1
24. 24.What is the remainder when 199 is divided by 3? (a) 2 (b) 1 (c) 0
25. 25.What is the remainder when 694 is divided by 12? (a) 9 (b) 2 (c) 11 (d) 10
26. 26.What is the remainder when 421 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
27. 27.Which of these numbers is divisible by 4? (a) 409 (b) 405 (c) 404 (d) 401
28. 28.What is the remainder when 506 is divided by 7? (a) 2 (b) 5 (c) 3 (d) 1
29. 29.What is the units digit of 8^44? (a) 2 (b) 6 (c) 4 (d) 8
30. 30.What is the remainder when 623 is divided by 9? (a) 7 (b) 1 (c) 2 (d) 3
31. 31.What is the units digit of 4^56? (a) 6 (b) 5 (c) 7 (d) 4
32. 32.What is the remainder when 548 is divided by 9? (a) 8 (b) 1 (c) 7 (d) 6
33. 33.Which of these numbers is divisible by 8? (a) 557 (b) 552 (c) 545 (d) 562
34. 34.What is the remainder when 290 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
35. 35.What is the units digit of 3^43? (a) 1 (b) 7 (c) 3 (d) 9
36. 36.What is the units digit of 8^25? (a) 8 (b) 2 (c) 4 (d) 6
37. 37.What is the units digit of 4^34? (a) 6 (b) 7 (c) 4 (d) 5
38. 38.What is the smallest number you must add to 89 to make it divisible by 8? (a) 1 (b) 5 (c) 6 (d) 7
39. 39.Which of these numbers is divisible by 3? (a) 197 (b) 203 (c) 200 (d) 198
40. 40.What is the smallest number you must add to 210 to make it divisible by 8? (a) 5 (b) 7 (c) 6 (d) 2

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