Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 570 · 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 units digit of 9^22? (a) 1 (b) 9 (c) 2 (d) 0
2. 2.What is the units digit of 8^47? (a) 2 (b) 8 (c) 6 (d) 4
3. 3.Which of these numbers is divisible by 8? (a) 2022 (b) 2011 (c) 2028 (d) 2016
4. 4.What is the units digit of 3^54? (a) 7 (b) 3 (c) 1 (d) 9
5. 5.What is the units digit of 4^53? (a) 6 (b) 4 (c) 3 (d) 5
6. 6.What is the remainder when 683 is divided by 11? (a) 1 (b) 2 (c) 10 (d) 0
7. 7.What is the smallest number you must add to 167 to make it divisible by 6? (a) 1 (b) 3 (c) 5 (d) 2
8. 8.What is the smallest number you must add to 139 to make it divisible by 3? (a) 2 (b) 1
9. 9.What is the smallest number you must add to 206 to make it divisible by 6? (a) 5 (b) 2 (c) 4 (d) 3
10. 10.What is the units digit of 7^29? (a) 7 (b) 3 (c) 1 (d) 9
11. 11.Which of these numbers is divisible by 8? (a) 408 (b) 410 (c) 422 (d) 407
12. 12.What is the smallest number you must add to 206 to make it divisible by 12? (a) 2 (b) 11 (c) 10 (d) 9
13. 13.What is the units digit of 8^58? (a) 2 (b) 4 (c) 6 (d) 8
14. 14.What is the remainder when 284 is divided by 9? (a) 5 (b) 6 (c) 4 (d) 7
15. 15.Which of these numbers is divisible by 9? (a) 2457 (b) 2453 (c) 2462 (d) 2469
16. 16.What is the remainder when 160 is divided by 9? (a) 6 (b) 7 (c) 2 (d) 8
17. 17.What is the remainder when 380 is divided by 6? (a) 1 (b) 2 (c) 4 (d) 3
18. 18.What is the units digit of 7^18? (a) 7 (b) 3 (c) 1 (d) 9
19. 19.Which of these numbers is divisible by 8? (a) 488 (b) 492 (c) 482 (d) 501
20. 20.What is the remainder when 471 is divided by 9? (a) 4 (b) 3 (c) 6 (d) 2
21. 21.What is the smallest number you must add to 295 to make it divisible by 4? (a) 2 (b) 3 (c) 1
22. 22.What is the remainder when 131 is divided by 12? (a) 1 (b) 10 (c) 11 (d) 9
23. 23.Which of these numbers is divisible by 3? (a) 245 (b) 241 (c) 248 (d) 243
24. 24.What is the smallest number you must add to 154 to make it divisible by 9? (a) 1 (b) 7 (c) 8 (d) 6
25. 25.What is the units digit of 2^20? (a) 4 (b) 6 (c) 8 (d) 2
26. 26.What is the smallest number you must add to 233 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
27. 27.What is the units digit of 9^30? (a) 9 (b) 1 (c) 0 (d) 2
28. 28.Which of these numbers is divisible by 8? (a) 840 (b) 853 (c) 837 (d) 843
29. 29.Which of these numbers is divisible by 11? (a) 6887 (b) 6878 (c) 6886 (d) 6904
30. 30.Which of these numbers is divisible by 8? (a) 4367 (b) 4355 (c) 4352 (d) 4350
31. 31.Which of these numbers is divisible by 6? (a) 1575 (b) 1589 (c) 1578 (d) 1581
32. 32.What is the units digit of 7^24? (a) 7 (b) 9 (c) 3 (d) 1
33. 33.What is the units digit of 8^30? (a) 8 (b) 2 (c) 4 (d) 6
34. 34.What is the remainder when 580 is divided by 11? (a) 7 (b) 3 (c) 9 (d) 8
35. 35.What is the units digit of 8^18? (a) 2 (b) 4 (c) 6 (d) 8
36. 36.Which of these numbers is divisible by 9? (a) 4010 (b) 4028 (c) 4020 (d) 4014
37. 37.What is the units digit of 2^10? (a) 4 (b) 6 (c) 2 (d) 8
38. 38.What is the remainder when 266 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
39. 39.What is the remainder when 206 is divided by 7? (a) 3 (b) 2 (c) 4 (d) 5
40. 40.What is the units digit of 7^45? (a) 9 (b) 3 (c) 1 (d) 7

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