Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 447 · 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 203 is divided by 11? (a) 6 (b) 7 (c) 4 (d) 5
2. 2.Which of these numbers is divisible by 3? (a) 553 (b) 559 (c) 555 (d) 556
3. 3.What is the remainder when 388 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 2
4. 4.Which of these numbers is divisible by 8? (a) 5368 (b) 5361 (c) 5377 (d) 5372
5. 5.What is the units digit of 7^46? (a) 3 (b) 1 (c) 7 (d) 9
6. 6.What is the remainder when 254 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
7. 7.What is the remainder when 237 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
8. 8.Which of these numbers is divisible by 8? (a) 6192 (b) 6190 (c) 6197 (d) 6204
9. 9.What is the units digit of 2^27? (a) 8 (b) 6 (c) 4 (d) 2
10. 10.Which of these numbers is divisible by 6? (a) 2828 (b) 2836 (c) 2840 (d) 2832
11. 11.What is the units digit of 3^12? (a) 3 (b) 7 (c) 1 (d) 9
12. 12.What is the remainder when 38 is divided by 3? (a) 2 (b) 0 (c) 1
13. 13.Which of these numbers is divisible by 3? (a) 518 (b) 511 (c) 513 (d) 514
14. 14.What is the remainder when 127 is divided by 3? (a) 1 (b) 0 (c) 2
15. 15.Which of these numbers is divisible by 3? (a) 1148 (b) 1151 (c) 1145 (d) 1146
16. 16.Which of these numbers is divisible by 3? (a) 458 (b) 455 (c) 461 (d) 456
17. 17.Which of these numbers is divisible by 8? (a) 1292 (b) 1275 (c) 1280 (d) 1287
18. 18.Which of these numbers is divisible by 3? (a) 1109 (b) 1106 (c) 1102 (d) 1104
19. 19.What is the units digit of 3^23? (a) 3 (b) 7 (c) 9 (d) 1
20. 20.What is the smallest number you must add to 744 to make it divisible by 10? (a) 6 (b) 7 (c) 5 (d) 4
21. 21.What is the smallest number you must add to 34 to make it divisible by 3? (a) 2 (b) 1
22. 22.What is the remainder when 157 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
23. 23.What is the smallest number you must add to 89 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
24. 24.Which of these numbers is divisible by 6? (a) 5050 (b) 5038 (c) 5045 (d) 5040
25. 25.What is the smallest number you must add to 143 to make it divisible by 6? (a) 5 (b) 2 (c) 1 (d) 3
26. 26.What is the smallest number you must add to 281 to make it divisible by 9? (a) 6 (b) 8 (c) 7 (d) 2
27. 27.What is the smallest number you must add to 61 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
28. 28.What is the smallest number you must add to 298 to make it divisible by 4? (a) 3 (b) 1 (c) 2
29. 29.What is the smallest number you must add to 456 to make it divisible by 7? (a) 5 (b) 6 (c) 1 (d) 4
30. 30.What is the units digit of 8^54? (a) 8 (b) 2 (c) 4 (d) 6
31. 31.What is the smallest number you must add to 116 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
32. 32.Which of these numbers is divisible by 9? (a) 1960 (b) 1965 (c) 1953 (d) 1946
33. 33.Which of these numbers is divisible by 8? (a) 4414 (b) 4430 (c) 4416 (d) 4421
34. 34.What is the smallest number you must add to 151 to make it divisible by 4? (a) 3 (b) 2 (c) 1
35. 35.What is the units digit of 8^47? (a) 2 (b) 4 (c) 8 (d) 6
36. 36.What is the units digit of 7^55? (a) 7 (b) 3 (c) 1 (d) 9
37. 37.What is the units digit of 4^34? (a) 5 (b) 6 (c) 4 (d) 7
38. 38.Which of these numbers is divisible by 3? (a) 2433 (b) 2434 (c) 2431 (d) 2438
39. 39.What is the units digit of 8^58? (a) 6 (b) 8 (c) 4 (d) 2
40. 40.What is the units digit of 9^27? (a) 1 (b) 9 (c) 8 (d) 7

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