Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 334 · 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 167 to make it divisible by 11? (a) 9 (b) 2 (c) 10 (d) 8
2. 2.Which of these numbers is divisible by 11? (a) 6217 (b) 6228 (c) 6226 (d) 6246
3. 3.What is the remainder when 222 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
4. 4.What is the remainder when 579 is divided by 8? (a) 4 (b) 5 (c) 2 (d) 3
5. 5.What is the units digit of 9^32? (a) 0 (b) 2 (c) 1 (d) 9
6. 6.Which of these numbers is divisible by 11? (a) 1090 (b) 1103 (c) 1084 (d) 1089
7. 7.What is the remainder when 198 is divided by 8? (a) 6 (b) 7 (c) 5 (d) 2
8. 8.What is the units digit of 3^13? (a) 1 (b) 9 (c) 7 (d) 3
9. 9.Which of these numbers is divisible by 11? (a) 869 (b) 886 (c) 874 (d) 867
10. 10.What is the smallest number you must add to 379 to make it divisible by 10? (a) 3 (b) 1 (c) 9 (d) 2
11. 11.Which of these numbers is divisible by 4? (a) 3352 (b) 3357 (c) 3353 (d) 3350
12. 12.What is the units digit of 4^29? (a) 5 (b) 4 (c) 3 (d) 6
13. 13.What is the remainder when 720 is divided by 11? (a) 7 (b) 5 (c) 4 (d) 6
14. 14.What is the remainder when 329 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 5
15. 15.What is the units digit of 8^22? (a) 2 (b) 6 (c) 8 (d) 4
16. 16.What is the units digit of 8^41? (a) 2 (b) 8 (c) 6 (d) 4
17. 17.What is the smallest number you must add to 484 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
18. 18.Which of these numbers is divisible by 6? (a) 2589 (b) 2580 (c) 2582 (d) 2577
19. 19.What is the smallest number you must add to 315 to make it divisible by 8? (a) 3 (b) 6 (c) 4 (d) 5
20. 20.Which of these numbers is divisible by 8? (a) 4867 (b) 4886 (c) 4875 (d) 4872
21. 21.What is the remainder when 187 is divided by 8? (a) 2 (b) 4 (c) 5 (d) 3
22. 22.What is the units digit of 9^57? (a) 9 (b) 1 (c) 8 (d) 7
23. 23.What is the smallest number you must add to 701 to make it divisible by 10? (a) 8 (b) 1 (c) 9 (d) 7
24. 24.What is the units digit of 8^39? (a) 4 (b) 2 (c) 8 (d) 6
25. 25.Which of these numbers is divisible by 11? (a) 4558 (b) 4569 (c) 4580 (d) 4565
26. 26.What is the remainder when 468 is divided by 10? (a) 2 (b) 8 (c) 9 (d) 7
27. 27.What is the smallest number you must add to 98 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
28. 28.What is the smallest number you must add to 673 to make it divisible by 11? (a) 10 (b) 9 (c) 2 (d) 8
29. 29.Which of these numbers is divisible by 3? (a) 1120 (b) 1117 (c) 1119 (d) 1124
30. 30.What is the remainder when 265 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
31. 31.What is the remainder when 719 is divided by 9? (a) 7 (b) 6 (c) 8 (d) 1
32. 32.Which of these numbers is divisible by 3? (a) 2636 (b) 2639 (c) 2637 (d) 2641
33. 33.Which of these numbers is divisible by 3? (a) 2357 (b) 2359 (c) 2353 (d) 2355
34. 34.What is the units digit of 8^56? (a) 4 (b) 8 (c) 6 (d) 2
35. 35.What is the smallest number you must add to 118 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
36. 36.What is the units digit of 9^59? (a) 1 (b) 8 (c) 7 (d) 9
37. 37.Which of these numbers is divisible by 8? (a) 883 (b) 900 (c) 888 (d) 890
38. 38.What is the smallest number you must add to 92 to make it divisible by 3? (a) 2 (b) 1
39. 39.What is the remainder when 131 is divided by 5? (a) 4 (b) 0 (c) 2 (d) 1
40. 40.Which of these numbers is divisible by 11? (a) 9405 (b) 9426 (c) 9398 (d) 9407

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