Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 423 · 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 8^49? (a) 6 (b) 8 (c) 2 (d) 4
2. 2.What is the remainder when 290 is divided by 12? (a) 10 (b) 1 (c) 2 (d) 3
3. 3.What is the smallest number you must add to 236 to make it divisible by 3? (a) 2 (b) 1
4. 4.What is the smallest number you must add to 85 to make it divisible by 4? (a) 1 (b) 2 (c) 3
5. 5.Which of these numbers is divisible by 8? (a) 2440 (b) 2435 (c) 2444 (d) 2451
6. 6.What is the smallest number you must add to 620 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
7. 7.What is the remainder when 264 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
8. 8.What is the smallest number you must add to 66 to make it divisible by 4? (a) 3 (b) 1 (c) 2
9. 9.What is the smallest number you must add to 479 to make it divisible by 6? (a) 3 (b) 1 (c) 5 (d) 2
10. 10.What is the units digit of 7^37? (a) 1 (b) 3 (c) 7 (d) 9
11. 11.What is the units digit of 3^14? (a) 3 (b) 9 (c) 1 (d) 7
12. 12.What is the remainder when 182 is divided by 3? (a) 1 (b) 2 (c) 0
13. 13.What is the units digit of 3^49? (a) 9 (b) 7 (c) 1 (d) 3
14. 14.What is the remainder when 396 is divided by 7? (a) 3 (b) 5 (c) 6 (d) 4
15. 15.What is the smallest number you must add to 324 to make it divisible by 7? (a) 2 (b) 4 (c) 6 (d) 5
16. 16.What is the remainder when 372 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
17. 17.Which of these numbers is divisible by 11? (a) 9779 (b) 9770 (c) 9781 (d) 9798
18. 18.What is the remainder when 396 is divided by 10? (a) 6 (b) 7 (c) 4 (d) 5
19. 19.What is the remainder when 133 is divided by 10? (a) 3 (b) 2 (c) 7 (d) 4
20. 20.What is the units digit of 3^13? (a) 3 (b) 1 (c) 7 (d) 9
21. 21.What is the units digit of 3^17? (a) 7 (b) 3 (c) 1 (d) 9
22. 22.What is the smallest number you must add to 311 to make it divisible by 8? (a) 2 (b) 1 (c) 7 (d) 3
23. 23.What is the smallest number you must add to 116 to make it divisible by 3? (a) 1 (b) 2
24. 24.What is the units digit of 3^53? (a) 3 (b) 1 (c) 7 (d) 9
25. 25.What is the smallest number you must add to 352 to make it divisible by 6? (a) 4 (b) 2 (c) 1 (d) 3
26. 26.What is the units digit of 3^25? (a) 1 (b) 3 (c) 7 (d) 9
27. 27.What is the units digit of 2^25? (a) 8 (b) 2 (c) 6 (d) 4
28. 28.What is the remainder when 122 is divided by 6? (a) 1 (b) 4 (c) 2 (d) 3
29. 29.Which of these numbers is divisible by 4? (a) 465 (b) 462 (c) 457 (d) 460
30. 30.What is the smallest number you must add to 446 to make it divisible by 10? (a) 6 (b) 3 (c) 4 (d) 5
31. 31.What is the smallest number you must add to 508 to make it divisible by 10? (a) 1 (b) 3 (c) 2 (d) 8
32. 32.What is the units digit of 8^22? (a) 2 (b) 8 (c) 4 (d) 6
33. 33.What is the remainder when 223 is divided by 8? (a) 1 (b) 5 (c) 7 (d) 6
34. 34.What is the units digit of 2^44? (a) 2 (b) 8 (c) 6 (d) 4
35. 35.What is the remainder when 182 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
36. 36.What is the units digit of 2^27? (a) 8 (b) 4 (c) 2 (d) 6
37. 37.Which of these numbers is divisible by 6? (a) 1398 (b) 1401 (c) 1393 (d) 1407
38. 38.What is the units digit of 3^21? (a) 9 (b) 7 (c) 1 (d) 3
39. 39.Which of these numbers is divisible by 3? (a) 1081 (b) 1084 (c) 1079 (d) 1080
40. 40.What is the units digit of 9^46? (a) 2 (b) 1 (c) 9 (d) 0

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