Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 525 · 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 8^11? (a) 8 (b) 2 (c) 6 (d) 4
2. 2.What is the units digit of 4^12? (a) 7 (b) 6 (c) 5 (d) 4
3. 3.Which of these numbers is divisible by 11? (a) 4243 (b) 4246 (c) 4256 (d) 4267
4. 4.What is the units digit of 8^36? (a) 6 (b) 8 (c) 2 (d) 4
5. 5.What is the units digit of 3^45? (a) 9 (b) 7 (c) 1 (d) 3
6. 6.What is the units digit of 8^13? (a) 4 (b) 6 (c) 8 (d) 2
7. 7.What is the remainder when 108 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
8. 8.What is the smallest number you must add to 122 to make it divisible by 4? (a) 3 (b) 1 (c) 2
9. 9.What is the units digit of 4^52? (a) 6 (b) 7 (c) 4 (d) 5
10. 10.What is the smallest number you must add to 238 to make it divisible by 3? (a) 1 (b) 2
11. 11.What is the remainder when 161 is divided by 10? (a) 0 (b) 1 (c) 2 (d) 9
12. 12.What is the units digit of 7^15? (a) 9 (b) 3 (c) 1 (d) 7
13. 13.Which of these numbers is divisible by 4? (a) 2285 (b) 2280 (c) 2283 (d) 2279
14. 14.What is the units digit of 7^42? (a) 9 (b) 1 (c) 3 (d) 7
15. 15.What is the remainder when 200 is divided by 6? (a) 3 (b) 1 (c) 2 (d) 4
16. 16.Which of these numbers is divisible by 4? (a) 2543 (b) 2534 (c) 2536 (d) 2537
17. 17.Which of these numbers is divisible by 11? (a) 1858 (b) 1868 (c) 1876 (d) 1859
18. 18.What is the smallest number you must add to 728 to make it divisible by 11? (a) 2 (b) 10 (c) 8 (d) 9
19. 19.What is the smallest number you must add to 134 to make it divisible by 10? (a) 7 (b) 5 (c) 4 (d) 6
20. 20.What is the smallest number you must add to 551 to make it divisible by 7? (a) 2 (b) 1 (c) 3 (d) 5
21. 21.What is the remainder when 589 is divided by 11? (a) 7 (b) 8 (c) 5 (d) 6
22. 22.Which of these numbers is divisible by 8? (a) 1666 (b) 1659 (c) 1673 (d) 1664
23. 23.Which of these numbers is divisible by 9? (a) 2457 (b) 2470 (c) 2454 (d) 2459
24. 24.What is the units digit of 8^26? (a) 2 (b) 6 (c) 8 (d) 4
25. 25.What is the smallest number you must add to 487 to make it divisible by 12? (a) 4 (b) 7 (c) 6 (d) 5
26. 26.What is the smallest number you must add to 282 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
27. 27.What is the remainder when 661 is divided by 10? (a) 2 (b) 9 (c) 1 (d) 0
28. 28.What is the remainder when 166 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
29. 29.Which of these numbers is divisible by 9? (a) 3930 (b) 3934 (c) 3933 (d) 3950
30. 30.Which of these numbers is divisible by 9? (a) 4970 (b) 4977 (c) 4988 (d) 4980
31. 31.What is the units digit of 4^16? (a) 7 (b) 5 (c) 4 (d) 6
32. 32.What is the remainder when 194 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 2
33. 33.Which of these numbers is divisible by 4? (a) 2468 (b) 2471 (c) 2467 (d) 2475
34. 34.Which of these numbers is divisible by 8? (a) 1190 (b) 1176 (c) 1170 (d) 1178
35. 35.What is the units digit of 3^56? (a) 1 (b) 9 (c) 3 (d) 7
36. 36.Which of these numbers is divisible by 9? (a) 1179 (b) 1171 (c) 1182 (d) 1196
37. 37.What is the units digit of 2^20? (a) 2 (b) 4 (c) 8 (d) 6
38. 38.Which of these numbers is divisible by 6? (a) 4262 (b) 4257 (c) 4254 (d) 4251
39. 39.What is the units digit of 8^28? (a) 8 (b) 4 (c) 2 (d) 6
40. 40.Which of these numbers is divisible by 3? (a) 1978 (b) 1977 (c) 1982 (d) 1976

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