Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 148 · 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 remainder when 397 is divided by 9? (a) 2 (b) 8 (c) 0 (d) 1
2. 2.Which of these numbers is divisible by 4? (a) 2062 (b) 2058 (c) 2056 (d) 2053
3. 3.Which of these numbers is divisible by 8? (a) 7086 (b) 7076 (c) 7070 (d) 7072
4. 4.Which of these numbers is divisible by 4? (a) 735 (b) 737 (c) 741 (d) 736
5. 5.What is the smallest number you must add to 158 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
6. 6.What is the smallest number you must add to 313 to make it divisible by 4? (a) 2 (b) 3 (c) 1
7. 7.What is the smallest number you must add to 764 to make it divisible by 12? (a) 8 (b) 5 (c) 4 (d) 3
8. 8.Which of these numbers is divisible by 3? (a) 1189 (b) 1191 (c) 1196 (d) 1192
9. 9.Which of these numbers is divisible by 6? (a) 2270 (b) 2279 (c) 2267 (d) 2268
10. 10.What is the remainder when 208 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 2
11. 11.What is the units digit of 2^46? (a) 2 (b) 6 (c) 8 (d) 4
12. 12.What is the units digit of 3^47? (a) 3 (b) 7 (c) 1 (d) 9
13. 13.What is the units digit of 3^56? (a) 3 (b) 1 (c) 7 (d) 9
14. 14.What is the remainder when 231 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 0
15. 15.What is the smallest number you must add to 631 to make it divisible by 8? (a) 7 (b) 2 (c) 3 (d) 1
16. 16.What is the smallest number you must add to 495 to make it divisible by 12? (a) 10 (b) 8 (c) 3 (d) 9
17. 17.What is the units digit of 3^52? (a) 9 (b) 3 (c) 7 (d) 1
18. 18.What is the units digit of 3^58? (a) 7 (b) 1 (c) 9 (d) 3
19. 19.What is the units digit of 9^32? (a) 9 (b) 0 (c) 1 (d) 2
20. 20.What is the units digit of 4^44? (a) 6 (b) 5 (c) 7 (d) 4
21. 21.Which of these numbers is divisible by 4? (a) 1232 (b) 1239 (c) 1231 (d) 1235
22. 22.What is the units digit of 8^31? (a) 2 (b) 4 (c) 8 (d) 6
23. 23.What is the remainder when 141 is divided by 10? (a) 9 (b) 0 (c) 1 (d) 2
24. 24.What is the units digit of 3^57? (a) 7 (b) 3 (c) 1 (d) 9
25. 25.What is the smallest number you must add to 783 to make it divisible by 12? (a) 9 (b) 3 (c) 10 (d) 8
26. 26.What is the remainder when 134 is divided by 12? (a) 3 (b) 10 (c) 1 (d) 2
27. 27.What is the remainder when 392 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
28. 28.What is the smallest number you must add to 132 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
29. 29.What is the units digit of 7^10? (a) 9 (b) 3 (c) 1 (d) 7
30. 30.What is the units digit of 2^25? (a) 6 (b) 4 (c) 8 (d) 2
31. 31.What is the units digit of 8^37? (a) 8 (b) 2 (c) 6 (d) 4
32. 32.What is the units digit of 2^27? (a) 4 (b) 8 (c) 2 (d) 6
33. 33.Which of these numbers is divisible by 9? (a) 3456 (b) 3464 (c) 3473 (d) 3453
34. 34.What is the smallest number you must add to 850 to make it divisible by 11? (a) 3 (b) 8 (c) 7 (d) 9
35. 35.What is the remainder when 544 is divided by 11? (a) 5 (b) 7 (c) 4 (d) 6
36. 36.What is the remainder when 156 is divided by 5? (a) 2 (b) 0 (c) 1 (d) 4
37. 37.Which of these numbers is divisible by 11? (a) 4905 (b) 4890 (c) 4895 (d) 4910
38. 38.What is the units digit of 9^48? (a) 9 (b) 1 (c) 2 (d) 0
39. 39.What is the smallest number you must add to 272 to make it divisible by 10? (a) 8 (b) 7 (c) 2 (d) 9
40. 40.What is the remainder when 110 is divided by 3? (a) 1 (b) 0 (c) 2

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