Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 512 · 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 11? (a) 5137 (b) 5154 (c) 5131 (d) 5139
2. 2.What is the remainder when 101 is divided by 10? (a) 9 (b) 1 (c) 2 (d) 0
3. 3.Which of these numbers is divisible by 11? (a) 7133 (b) 7128 (c) 7123 (d) 7141
4. 4.What is the remainder when 307 is divided by 6? (a) 2 (b) 5 (c) 1 (d) 0
5. 5.What is the units digit of 4^13? (a) 6 (b) 4 (c) 5 (d) 3
6. 6.What is the smallest number you must add to 427 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 1
7. 7.What is the smallest number you must add to 906 to make it divisible by 12? (a) 5 (b) 7 (c) 8 (d) 6
8. 8.What is the units digit of 4^15? (a) 3 (b) 5 (c) 6 (d) 4
9. 9.What is the units digit of 4^38? (a) 7 (b) 4 (c) 5 (d) 6
10. 10.Which of these numbers is divisible by 11? (a) 519 (b) 546 (c) 528 (d) 531
11. 11.Which of these numbers is divisible by 6? (a) 1586 (b) 1591 (c) 1584 (d) 1582
12. 12.What is the units digit of 7^42? (a) 1 (b) 3 (c) 9 (d) 7
13. 13.What is the units digit of 9^35? (a) 8 (b) 9 (c) 1 (d) 7
14. 14.What is the units digit of 4^53? (a) 4 (b) 5 (c) 6 (d) 3
15. 15.What is the smallest number you must add to 884 to make it divisible by 11? (a) 4 (b) 8 (c) 7 (d) 6
16. 16.What is the units digit of 8^34? (a) 2 (b) 8 (c) 6 (d) 4
17. 17.What is the remainder when 301 is divided by 6? (a) 0 (b) 5 (c) 1 (d) 2
18. 18.What is the remainder when 476 is divided by 9? (a) 1 (b) 8 (c) 6 (d) 7
19. 19.What is the smallest number you must add to 341 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
20. 20.What is the smallest number you must add to 232 to make it divisible by 3? (a) 2 (b) 1
21. 21.What is the units digit of 7^34? (a) 3 (b) 9 (c) 7 (d) 1
22. 22.What is the smallest number you must add to 656 to make it divisible by 11? (a) 3 (b) 7 (c) 5 (d) 4
23. 23.Which of these numbers is divisible by 9? (a) 6003 (b) 6004 (c) 6013 (d) 6001
24. 24.What is the units digit of 9^10? (a) 1 (b) 9 (c) 0 (d) 2
25. 25.Which of these numbers is divisible by 4? (a) 997 (b) 1002 (c) 996 (d) 994
26. 26.What is the smallest number you must add to 437 to make it divisible by 9? (a) 4 (b) 3 (c) 6 (d) 5
27. 27.What is the units digit of 8^10? (a) 2 (b) 6 (c) 8 (d) 4
28. 28.What is the smallest number you must add to 190 to make it divisible by 8? (a) 2 (b) 3 (c) 6 (d) 1
29. 29.What is the remainder when 186 is divided by 5? (a) 2 (b) 1 (c) 0 (d) 4
30. 30.What is the smallest number you must add to 302 to make it divisible by 4? (a) 1 (b) 2 (c) 3
31. 31.Which of these numbers is divisible by 4? (a) 2687 (b) 2691 (c) 2683 (d) 2684
32. 32.What is the smallest number you must add to 516 to make it divisible by 7? (a) 1 (b) 2 (c) 5 (d) 3
33. 33.What is the units digit of 2^41? (a) 8 (b) 4 (c) 2 (d) 6
34. 34.Which of these numbers is divisible by 8? (a) 2644 (b) 2638 (c) 2640 (d) 2649
35. 35.Which of these numbers is divisible by 9? (a) 591 (b) 596 (c) 608 (d) 594
36. 36.What is the remainder when 254 is divided by 8? (a) 5 (b) 7 (c) 2 (d) 6
37. 37.What is the units digit of 7^37? (a) 1 (b) 9 (c) 3 (d) 7
38. 38.What is the smallest number you must add to 115 to make it divisible by 10? (a) 7 (b) 6 (c) 4 (d) 5
39. 39.What is the remainder when 292 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 2
40. 40.Which of these numbers is divisible by 3? (a) 2415 (b) 2420 (c) 2414 (d) 2417

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