Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 586 · 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 431 is divided by 9? (a) 6 (b) 7 (c) 1 (d) 8
2. 2.Which of these numbers is divisible by 4? (a) 2837 (b) 2841 (c) 2836 (d) 2835
3. 3.Which of these numbers is divisible by 8? (a) 3363 (b) 3375 (c) 3360 (d) 3358
4. 4.What is the remainder when 199 is divided by 6? (a) 2 (b) 5 (c) 0 (d) 1
5. 5.What is the remainder when 211 is divided by 3? (a) 2 (b) 0 (c) 1
6. 6.What is the smallest number you must add to 276 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
7. 7.What is the units digit of 4^55? (a) 6 (b) 5 (c) 3 (d) 4
8. 8.Which of these numbers is divisible by 4? (a) 1027 (b) 1024 (c) 1022 (d) 1030
9. 9.What is the remainder when 211 is divided by 6? (a) 0 (b) 1 (c) 5 (d) 2
10. 10.What is the units digit of 2^60? (a) 8 (b) 6 (c) 4 (d) 2
11. 11.Which of these numbers is divisible by 6? (a) 1356 (b) 1364 (c) 1359 (d) 1355
12. 12.What is the units digit of 8^24? (a) 4 (b) 8 (c) 2 (d) 6
13. 13.Which of these numbers is divisible by 8? (a) 1777 (b) 1784 (c) 1795 (d) 1789
14. 14.Which of these numbers is divisible by 4? (a) 3270 (b) 3274 (c) 3272 (d) 3277
15. 15.What is the remainder when 253 is divided by 9? (a) 2 (b) 0 (c) 8 (d) 1
16. 16.What is the units digit of 7^54? (a) 1 (b) 3 (c) 9 (d) 7
17. 17.What is the remainder when 428 is divided by 9? (a) 4 (b) 6 (c) 5 (d) 7
18. 18.What is the units digit of 7^38? (a) 3 (b) 9 (c) 1 (d) 7
19. 19.What is the remainder when 62 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 1
20. 20.What is the units digit of 4^22? (a) 5 (b) 7 (c) 4 (d) 6
21. 21.What is the smallest number you must add to 98 to make it divisible by 9? (a) 8 (b) 2 (c) 1 (d) 3
22. 22.What is the remainder when 776 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 7
23. 23.What is the units digit of 4^17? (a) 6 (b) 4 (c) 5 (d) 3
24. 24.Which of these numbers is divisible by 4? (a) 3499 (b) 3495 (c) 3496 (d) 3501
25. 25.What is the smallest number you must add to 65 to make it divisible by 3? (a) 2 (b) 1
26. 26.What is the remainder when 293 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
27. 27.What is the units digit of 2^18? (a) 2 (b) 4 (c) 8 (d) 6
28. 28.Which of these numbers is divisible by 4? (a) 1078 (b) 1080 (c) 1086 (d) 1081
29. 29.Which of these numbers is divisible by 3? (a) 629 (b) 623 (c) 626 (d) 624
30. 30.What is the remainder when 916 is divided by 12? (a) 8 (b) 4 (c) 5 (d) 3
31. 31.What is the remainder when 241 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
32. 32.What is the smallest number you must add to 272 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
33. 33.What is the smallest number you must add to 146 to make it divisible by 4? (a) 1 (b) 3 (c) 2
34. 34.What is the smallest number you must add to 647 to make it divisible by 10? (a) 7 (b) 4 (c) 2 (d) 3
35. 35.Which of these numbers is divisible by 3? (a) 1929 (b) 1931 (c) 1933 (d) 1927
36. 36.What is the units digit of 9^44? (a) 9 (b) 2 (c) 1 (d) 0
37. 37.What is the smallest number you must add to 249 to make it divisible by 10? (a) 9 (b) 2 (c) 1 (d) 3
38. 38.What is the units digit of 3^16? (a) 3 (b) 7 (c) 1 (d) 9
39. 39.What is the smallest number you must add to 400 to make it divisible by 9? (a) 4 (b) 7 (c) 5 (d) 6
40. 40.What is the remainder when 146 is divided by 9? (a) 3 (b) 7 (c) 2 (d) 1

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