Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 522 · 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 6? (a) 2994 (b) 2995 (c) 3001 (d) 2993
2. 2.Which of these numbers is divisible by 3? (a) 288 (b) 292 (c) 286 (d) 289
3. 3.What is the remainder when 296 is divided by 9? (a) 8 (b) 7 (c) 6 (d) 1
4. 4.What is the smallest number you must add to 611 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
5. 5.Which of these numbers is divisible by 9? (a) 6570 (b) 6586 (c) 6577 (d) 6566
6. 6.What is the remainder when 170 is divided by 3? (a) 1 (b) 2 (c) 0
7. 7.What is the remainder when 285 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 2
8. 8.Which of these numbers is divisible by 11? (a) 5403 (b) 5390 (c) 5387 (d) 5394
9. 9.What is the remainder when 367 is divided by 10? (a) 6 (b) 7 (c) 8 (d) 3
10. 10.Which of these numbers is divisible by 4? (a) 1845 (b) 1838 (c) 1841 (d) 1840
11. 11.Which of these numbers is divisible by 9? (a) 2549 (b) 2547 (c) 2539 (d) 2560
12. 12.Which of these numbers is divisible by 6? (a) 2553 (b) 2557 (c) 2566 (d) 2556
13. 13.Which of these numbers is divisible by 6? (a) 3702 (b) 3701 (c) 3705 (d) 3710
14. 14.What is the units digit of 7^41? (a) 7 (b) 3 (c) 9 (d) 1
15. 15.What is the units digit of 8^14? (a) 2 (b) 4 (c) 8 (d) 6
16. 16.Which of these numbers is divisible by 4? (a) 3102 (b) 3104 (c) 3109 (d) 3107
17. 17.Which of these numbers is divisible by 8? (a) 1058 (b) 1064 (c) 1079 (d) 1065
18. 18.What is the remainder when 59 is divided by 3? (a) 2 (b) 1 (c) 0
19. 19.What is the units digit of 2^40? (a) 2 (b) 4 (c) 6 (d) 8
20. 20.What is the remainder when 514 is divided by 9? (a) 1 (b) 0 (c) 8 (d) 2
21. 21.What is the smallest number you must add to 194 to make it divisible by 3? (a) 2 (b) 1
22. 22.Which of these numbers is divisible by 11? (a) 3926 (b) 3902 (c) 3914 (d) 3905
23. 23.What is the smallest number you must add to 94 to make it divisible by 4? (a) 1 (b) 3 (c) 2
24. 24.Which of these numbers is divisible by 3? (a) 919 (b) 916 (c) 915 (d) 913
25. 25.What is the remainder when 546 is divided by 9? (a) 6 (b) 7 (c) 5 (d) 3
26. 26.Which of these numbers is divisible by 9? (a) 2664 (b) 2676 (c) 2663 (d) 2666
27. 27.What is the smallest number you must add to 401 to make it divisible by 8? (a) 5 (b) 7 (c) 1 (d) 6
28. 28.Which of these numbers is divisible by 4? (a) 2750 (b) 2752 (c) 2755 (d) 2759
29. 29.What is the smallest number you must add to 359 to make it divisible by 8? (a) 7 (b) 1 (c) 2 (d) 3
30. 30.What is the units digit of 3^60? (a) 9 (b) 3 (c) 1 (d) 7
31. 31.What is the units digit of 3^25? (a) 9 (b) 7 (c) 3 (d) 1
32. 32.What is the units digit of 2^18? (a) 2 (b) 4 (c) 8 (d) 6
33. 33.What is the units digit of 9^19? (a) 8 (b) 7 (c) 1 (d) 9
34. 34.What is the remainder when 95 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 5
35. 35.Which of these numbers is divisible by 6? (a) 4411 (b) 4404 (c) 4409 (d) 4402
36. 36.Which of these numbers is divisible by 3? (a) 914 (b) 912 (c) 911 (d) 916
37. 37.What is the units digit of 3^36? (a) 7 (b) 3 (c) 9 (d) 1
38. 38.What is the smallest number you must add to 561 to make it divisible by 10? (a) 7 (b) 8 (c) 9 (d) 1
39. 39.What is the units digit of 9^17? (a) 1 (b) 9 (c) 7 (d) 8
40. 40.What is the smallest number you must add to 448 to make it divisible by 11? (a) 3 (b) 8 (c) 4 (d) 2

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