Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 487 · 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 464 to make it divisible by 10? (a) 7 (b) 5 (c) 4 (d) 6
2. 2.Which of these numbers is divisible by 4? (a) 627 (b) 628 (c) 630 (d) 634
3. 3.Which of these numbers is divisible by 8? (a) 3703 (b) 3695 (c) 3707 (d) 3696
4. 4.Which of these numbers is divisible by 9? (a) 5452 (b) 5461 (c) 5470 (d) 5454
5. 5.What is the smallest number you must add to 445 to make it divisible by 12? (a) 10 (b) 9 (c) 11 (d) 1
6. 6.What is the smallest number you must add to 418 to make it divisible by 8? (a) 7 (b) 5 (c) 6 (d) 2
7. 7.Which of these numbers is divisible by 9? (a) 679 (b) 667 (c) 675 (d) 686
8. 8.What is the units digit of 4^41? (a) 3 (b) 6 (c) 4 (d) 5
9. 9.What is the remainder when 606 is divided by 8? (a) 5 (b) 6 (c) 7 (d) 2
10. 10.What is the remainder when 294 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
11. 11.What is the remainder when 70 is divided by 3? (a) 0 (b) 2 (c) 1
12. 12.What is the units digit of 2^44? (a) 8 (b) 4 (c) 6 (d) 2
13. 13.What is the units digit of 9^27? (a) 7 (b) 1 (c) 9 (d) 8
14. 14.What is the remainder when 233 is divided by 12? (a) 4 (b) 5 (c) 6 (d) 7
15. 15.Which of these numbers is divisible by 11? (a) 1133 (b) 1137 (c) 1153 (d) 1129
16. 16.What is the units digit of 8^58? (a) 2 (b) 8 (c) 4 (d) 6
17. 17.What is the smallest number you must add to 375 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 7
18. 18.What is the remainder when 396 is divided by 10? (a) 7 (b) 6 (c) 5 (d) 4
19. 19.Which of these numbers is divisible by 4? (a) 886 (b) 894 (c) 888 (d) 891
20. 20.What is the remainder when 396 is divided by 8? (a) 5 (b) 3 (c) 6 (d) 4
21. 21.What is the smallest number you must add to 247 to make it divisible by 4? (a) 3 (b) 2 (c) 1
22. 22.What is the units digit of 2^17? (a) 6 (b) 2 (c) 4 (d) 8
23. 23.What is the units digit of 9^42? (a) 0 (b) 2 (c) 9 (d) 1
24. 24.What is the smallest number you must add to 321 to make it divisible by 4? (a) 3 (b) 2 (c) 1
25. 25.Which of these numbers is divisible by 8? (a) 3920 (b) 3925 (c) 3935 (d) 3918
26. 26.What is the remainder when 502 is divided by 10? (a) 1 (b) 3 (c) 2 (d) 8
27. 27.What is the units digit of 8^35? (a) 6 (b) 2 (c) 4 (d) 8
28. 28.What is the units digit of 2^14? (a) 2 (b) 4 (c) 8 (d) 6
29. 29.Which of these numbers is divisible by 8? (a) 5062 (b) 5066 (c) 5079 (d) 5064
30. 30.Which of these numbers is divisible by 11? (a) 572 (b) 569 (c) 590 (d) 575
31. 31.What is the smallest number you must add to 766 to make it divisible by 11? (a) 3 (b) 7 (c) 4 (d) 5
32. 32.What is the smallest number you must add to 481 to make it divisible by 12? (a) 10 (b) 9 (c) 1 (d) 11
33. 33.What is the units digit of 9^23? (a) 7 (b) 8 (c) 1 (d) 9
34. 34.What is the smallest number you must add to 197 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
35. 35.What is the smallest number you must add to 816 to make it divisible by 11? (a) 9 (b) 10 (c) 8 (d) 2
36. 36.What is the smallest number you must add to 326 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
37. 37.What is the remainder when 125 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
38. 38.Which of these numbers is divisible by 9? (a) 2568 (b) 2557 (c) 2581 (d) 2565
39. 39.What is the units digit of 2^55? (a) 2 (b) 6 (c) 4 (d) 8
40. 40.Which of these numbers is divisible by 4? (a) 2639 (b) 2643 (c) 2640 (d) 2646

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