Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 448 · 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 454 to make it divisible by 12? (a) 1 (b) 2 (c) 10 (d) 3
2. 2.What is the units digit of 7^41? (a) 7 (b) 9 (c) 3 (d) 1
3. 3.What is the smallest number you must add to 277 to make it divisible by 8? (a) 5 (b) 3 (c) 2 (d) 4
4. 4.Which of these numbers is divisible by 4? (a) 2311 (b) 2318 (c) 2312 (d) 2313
5. 5.Which of these numbers is divisible by 8? (a) 734 (b) 737 (c) 727 (d) 728
6. 6.Which of these numbers is divisible by 3? (a) 2265 (b) 2269 (c) 2264 (d) 2267
7. 7.What is the smallest number you must add to 101 to make it divisible by 8? (a) 2 (b) 3 (c) 4 (d) 5
8. 8.What is the units digit of 8^28? (a) 4 (b) 2 (c) 8 (d) 6
9. 9.What is the units digit of 4^25? (a) 6 (b) 3 (c) 5 (d) 4
10. 10.What is the units digit of 2^19? (a) 4 (b) 6 (c) 2 (d) 8
11. 11.Which of these numbers is divisible by 6? (a) 3825 (b) 3828 (c) 3830 (d) 3838
12. 12.What is the remainder when 195 is divided by 7? (a) 4 (b) 6 (c) 1 (d) 5
13. 13.Which of these numbers is divisible by 4? (a) 3323 (b) 3318 (c) 3326 (d) 3320
14. 14.What is the units digit of 4^30? (a) 6 (b) 4 (c) 5 (d) 7
15. 15.What is the units digit of 3^30? (a) 3 (b) 7 (c) 9 (d) 1
16. 16.Which of these numbers is divisible by 4? (a) 667 (b) 675 (c) 670 (d) 668
17. 17.What is the smallest number you must add to 378 to make it divisible by 8? (a) 7 (b) 6 (c) 2 (d) 5
18. 18.What is the units digit of 3^15? (a) 7 (b) 9 (c) 1 (d) 3
19. 19.What is the smallest number you must add to 38 to make it divisible by 3? (a) 2 (b) 1
20. 20.What is the units digit of 8^38? (a) 8 (b) 2 (c) 6 (d) 4
21. 21.What is the remainder when 468 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 1
22. 22.What is the smallest number you must add to 207 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
23. 23.What is the smallest number you must add to 191 to make it divisible by 4? (a) 2 (b) 3 (c) 1
24. 24.What is the smallest number you must add to 332 to make it divisible by 11? (a) 8 (b) 2 (c) 10 (d) 9
25. 25.Which of these numbers is divisible by 4? (a) 3405 (b) 3402 (c) 3398 (d) 3400
26. 26.Which of these numbers is divisible by 8? (a) 2485 (b) 2466 (c) 2472 (d) 2478
27. 27.What is the units digit of 3^11? (a) 7 (b) 1 (c) 9 (d) 3
28. 28.What is the smallest number you must add to 63 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
29. 29.What is the remainder when 196 is divided by 11? (a) 8 (b) 9 (c) 2 (d) 10
30. 30.What is the smallest number you must add to 885 to make it divisible by 12? (a) 9 (b) 4 (c) 3 (d) 2
31. 31.What is the remainder when 207 is divided by 10? (a) 3 (b) 6 (c) 8 (d) 7
32. 32.What is the remainder when 142 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
33. 33.What is the smallest number you must add to 52 to make it divisible by 3? (a) 1 (b) 2
34. 34.What is the remainder when 280 is divided by 12? (a) 3 (b) 4 (c) 8 (d) 5
35. 35.Which of these numbers is divisible by 9? (a) 7912 (b) 7932 (c) 7925 (d) 7920
36. 36.Which of these numbers is divisible by 11? (a) 7003 (b) 6993 (c) 7017 (d) 6996
37. 37.What is the smallest number you must add to 201 to make it divisible by 4? (a) 2 (b) 3 (c) 1
38. 38.What is the smallest number you must add to 242 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
39. 39.What is the remainder when 186 is divided by 9? (a) 5 (b) 7 (c) 3 (d) 6
40. 40.What is the units digit of 4^39? (a) 3 (b) 6 (c) 4 (d) 5

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