Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 533 · 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 3^54? (a) 9 (b) 1 (c) 3 (d) 7
2. 2.What is the remainder when 148 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 6
3. 3.What is the remainder when 512 is divided by 10? (a) 1 (b) 2 (c) 8 (d) 3
4. 4.What is the remainder when 317 is divided by 11? (a) 2 (b) 8 (c) 9 (d) 10
5. 5.What is the units digit of 7^58? (a) 9 (b) 7 (c) 1 (d) 3
6. 6.What is the smallest number you must add to 82 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
7. 7.What is the units digit of 2^22? (a) 6 (b) 4 (c) 2 (d) 8
8. 8.What is the smallest number you must add to 206 to make it divisible by 4? (a) 1 (b) 3 (c) 2
9. 9.What is the remainder when 291 is divided by 11? (a) 4 (b) 5 (c) 6 (d) 7
10. 10.What is the remainder when 175 is divided by 3? (a) 1 (b) 2 (c) 0
11. 11.What is the remainder when 497 is divided by 8? (a) 0 (b) 1 (c) 2 (d) 7
12. 12.What is the remainder when 157 is divided by 8? (a) 3 (b) 4 (c) 6 (d) 5
13. 13.What is the smallest number you must add to 127 to make it divisible by 3? (a) 1 (b) 2
14. 14.Which of these numbers is divisible by 11? (a) 2628 (b) 2607 (c) 2610 (d) 2601
15. 15.What is the remainder when 250 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
16. 16.What is the remainder when 188 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
17. 17.Which of these numbers is divisible by 4? (a) 2065 (b) 2061 (c) 2071 (d) 2064
18. 18.What is the smallest number you must add to 104 to make it divisible by 7? (a) 1 (b) 2 (c) 3 (d) 6
19. 19.What is the smallest number you must add to 402 to make it divisible by 8? (a) 7 (b) 6 (c) 5 (d) 2
20. 20.What is the smallest number you must add to 241 to make it divisible by 12? (a) 11 (b) 1 (c) 10 (d) 9
21. 21.Which of these numbers is divisible by 6? (a) 636 (b) 632 (c) 646 (d) 640
22. 22.Which of these numbers is divisible by 8? (a) 1696 (b) 1691 (c) 1697 (d) 1705
23. 23.What is the remainder when 147 is divided by 12? (a) 2 (b) 3 (c) 9 (d) 4
24. 24.What is the remainder when 183 is divided by 7? (a) 0 (b) 2 (c) 6 (d) 1
25. 25.Which of these numbers is divisible by 6? (a) 4999 (b) 5014 (c) 5004 (d) 5007
26. 26.What is the units digit of 3^23? (a) 1 (b) 7 (c) 9 (d) 3
27. 27.What is the units digit of 8^51? (a) 6 (b) 8 (c) 2 (d) 4
28. 28.What is the remainder when 207 is divided by 7? (a) 3 (b) 5 (c) 4 (d) 6
29. 29.What is the units digit of 7^44? (a) 9 (b) 7 (c) 1 (d) 3
30. 30.Which of these numbers is divisible by 6? (a) 3681 (b) 3669 (c) 3672 (d) 3675
31. 31.What is the units digit of 8^21? (a) 8 (b) 2 (c) 4 (d) 6
32. 32.What is the units digit of 3^36? (a) 7 (b) 1 (c) 3 (d) 9
33. 33.What is the smallest number you must add to 173 to make it divisible by 3? (a) 1 (b) 2
34. 34.Which of these numbers is divisible by 3? (a) 2196 (b) 2197 (c) 2194 (d) 2201
35. 35.Which of these numbers is divisible by 11? (a) 4853 (b) 4866 (c) 4876 (d) 4862
36. 36.What is the units digit of 2^19? (a) 2 (b) 8 (c) 4 (d) 6
37. 37.Which of these numbers is divisible by 9? (a) 5788 (b) 5779 (c) 5787 (d) 5800
38. 38.What is the units digit of 8^25? (a) 8 (b) 4 (c) 2 (d) 6
39. 39.What is the remainder when 203 is divided by 3? (a) 2 (b) 0 (c) 1
40. 40.What is the smallest number you must add to 887 to make it divisible by 12? (a) 2 (b) 3 (c) 11 (d) 1

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