Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 523 · 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 4^29? (a) 3 (b) 5 (c) 4 (d) 6
2. 2.What is the units digit of 9^45? (a) 9 (b) 1 (c) 7 (d) 8
3. 3.What is the smallest number you must add to 226 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 2
4. 4.What is the remainder when 422 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 1
5. 5.What is the smallest number you must add to 76 to make it divisible by 6? (a) 1 (b) 2 (c) 4 (d) 3
6. 6.What is the units digit of 2^49? (a) 2 (b) 4 (c) 8 (d) 6
7. 7.Which of these numbers is divisible by 9? (a) 5935 (b) 5923 (c) 5948 (d) 5931
8. 8.Which of these numbers is divisible by 9? (a) 2684 (b) 2704 (c) 2693 (d) 2691
9. 9.What is the remainder when 286 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
10. 10.What is the remainder when 460 is divided by 7? (a) 6 (b) 2 (c) 4 (d) 5
11. 11.What is the remainder when 173 is divided by 10? (a) 4 (b) 7 (c) 2 (d) 3
12. 12.What is the smallest number you must add to 239 to make it divisible by 4? (a) 3 (b) 2 (c) 1
13. 13.Which of these numbers is divisible by 11? (a) 9470 (b) 9487 (c) 9471 (d) 9475
14. 14.Which of these numbers is divisible by 3? (a) 2297 (b) 2294 (c) 2295 (d) 2300
15. 15.What is the units digit of 4^41? (a) 3 (b) 5 (c) 4 (d) 6
16. 16.What is the smallest number you must add to 267 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
17. 17.What is the remainder when 241 is divided by 11? (a) 8 (b) 10 (c) 9 (d) 1
18. 18.What is the remainder when 137 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
19. 19.What is the remainder when 166 is divided by 3? (a) 1 (b) 2 (c) 0
20. 20.What is the remainder when 212 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
21. 21.Which of these numbers is divisible by 4? (a) 3419 (b) 3423 (c) 3427 (d) 3420
22. 22.What is the units digit of 2^23? (a) 6 (b) 4 (c) 8 (d) 2
23. 23.What is the remainder when 438 is divided by 7? (a) 5 (b) 4 (c) 3 (d) 6
24. 24.Which of these numbers is divisible by 6? (a) 4238 (b) 4226 (c) 4231 (d) 4230
25. 25.What is the units digit of 8^20? (a) 8 (b) 4 (c) 2 (d) 6
26. 26.What is the units digit of 8^12? (a) 4 (b) 8 (c) 2 (d) 6
27. 27.What is the smallest number you must add to 136 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 3
28. 28.What is the smallest number you must add to 508 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3
29. 29.Which of these numbers is divisible by 8? (a) 6656 (b) 6649 (c) 6662 (d) 6667
30. 30.What is the smallest number you must add to 157 to make it divisible by 3? (a) 1 (b) 2
31. 31.What is the remainder when 164 is divided by 12? (a) 9 (b) 8 (c) 7 (d) 4
32. 32.What is the remainder when 194 is divided by 12? (a) 10 (b) 3 (c) 1 (d) 2
33. 33.Which of these numbers is divisible by 4? (a) 2951 (b) 2954 (c) 2952 (d) 2958
34. 34.What is the remainder when 255 is divided by 6? (a) 5 (b) 3 (c) 2 (d) 4
35. 35.What is the units digit of 3^33? (a) 7 (b) 1 (c) 9 (d) 3
36. 36.Which of these numbers is divisible by 6? (a) 665 (b) 657 (c) 667 (d) 660
37. 37.What is the remainder when 339 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
38. 38.Which of these numbers is divisible by 11? (a) 2565 (b) 2563 (c) 2554 (d) 2575
39. 39.What is the smallest number you must add to 261 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
40. 40.What is the remainder when 395 is divided by 9? (a) 6 (b) 7 (c) 8 (d) 1

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