Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 131 · 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 9^25? (a) 8 (b) 1 (c) 9 (d) 7
2. 2.What is the units digit of 9^15? (a) 8 (b) 7 (c) 9 (d) 1
3. 3.What is the units digit of 3^36? (a) 3 (b) 7 (c) 9 (d) 1
4. 4.What is the smallest number you must add to 118 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
5. 5.What is the units digit of 7^22? (a) 3 (b) 1 (c) 9 (d) 7
6. 6.What is the smallest number you must add to 58 to make it divisible by 3? (a) 1 (b) 2
7. 7.Which of these numbers is divisible by 9? (a) 5080 (b) 5088 (c) 5095 (d) 5085
8. 8.What is the remainder when 189 is divided by 11? (a) 1 (b) 3 (c) 2 (d) 9
9. 9.What is the remainder when 459 is divided by 8? (a) 3 (b) 4 (c) 2 (d) 5
10. 10.What is the units digit of 4^17? (a) 6 (b) 3 (c) 4 (d) 5
11. 11.What is the units digit of 3^10? (a) 7 (b) 1 (c) 3 (d) 9
12. 12.What is the smallest number you must add to 56 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
13. 13.What is the remainder when 98 is divided by 6? (a) 4 (b) 1 (c) 2 (d) 3
14. 14.What is the units digit of 8^50? (a) 8 (b) 4 (c) 6 (d) 2
15. 15.What is the smallest number you must add to 134 to make it divisible by 7? (a) 5 (b) 6 (c) 1 (d) 4
16. 16.What is the remainder when 546 is divided by 11? (a) 7 (b) 8 (c) 6 (d) 4
17. 17.What is the units digit of 9^54? (a) 9 (b) 1 (c) 2 (d) 0
18. 18.What is the smallest number you must add to 257 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
19. 19.What is the remainder when 208 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
20. 20.What is the units digit of 9^13? (a) 9 (b) 7 (c) 8 (d) 1
21. 21.What is the smallest number you must add to 872 to make it divisible by 12? (a) 8 (b) 4 (c) 3 (d) 5
22. 22.What is the units digit of 9^16? (a) 2 (b) 0 (c) 9 (d) 1
23. 23.Which of these numbers is divisible by 4? (a) 883 (b) 874 (c) 876 (d) 879
24. 24.What is the units digit of 7^14? (a) 9 (b) 3 (c) 1 (d) 7
25. 25.Which of these numbers is divisible by 8? (a) 1913 (b) 1920 (c) 1931 (d) 1925
26. 26.What is the units digit of 9^14? (a) 0 (b) 9 (c) 2 (d) 1
27. 27.What is the remainder when 201 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
28. 28.What is the remainder when 225 is divided by 7? (a) 6 (b) 0 (c) 1 (d) 2
29. 29.What is the units digit of 8^59? (a) 4 (b) 8 (c) 2 (d) 6
30. 30.What is the units digit of 2^58? (a) 6 (b) 8 (c) 4 (d) 2
31. 31.What is the units digit of 2^15? (a) 6 (b) 8 (c) 4 (d) 2
32. 32.What is the smallest number you must add to 323 to make it divisible by 9? (a) 2 (b) 8 (c) 3 (d) 1
33. 33.What is the smallest number you must add to 143 to make it divisible by 6? (a) 1 (b) 2 (c) 5 (d) 3
34. 34.What is the remainder when 106 is divided by 3? (a) 1 (b) 2 (c) 0
35. 35.What is the units digit of 7^56? (a) 1 (b) 3 (c) 7 (d) 9
36. 36.What is the smallest number you must add to 294 to make it divisible by 12? (a) 8 (b) 7 (c) 5 (d) 6
37. 37.What is the smallest number you must add to 582 to make it divisible by 10? (a) 2 (b) 8 (c) 9 (d) 7
38. 38.What is the remainder when 176 is divided by 9? (a) 7 (b) 6 (c) 5 (d) 4
39. 39.What is the units digit of 3^11? (a) 7 (b) 1 (c) 3 (d) 9
40. 40.Which of these numbers is divisible by 4? (a) 359 (b) 361 (c) 354 (d) 356

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