Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 162 · 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 3^10? (a) 3 (b) 7 (c) 9 (d) 1
2. 2.What is the smallest number you must add to 436 to make it divisible by 7? (a) 2 (b) 6 (c) 5 (d) 4
3. 3.What is the smallest number you must add to 91 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the units digit of 4^59? (a) 3 (b) 6 (c) 5 (d) 4
5. 5.Which of these numbers is divisible by 11? (a) 6705 (b) 6695 (c) 6699 (d) 6712
6. 6.What is the remainder when 59 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
7. 7.What is the remainder when 57 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
8. 8.What is the units digit of 3^43? (a) 1 (b) 7 (c) 3 (d) 9
9. 9.Which of these numbers is divisible by 6? (a) 1243 (b) 1241 (c) 1251 (d) 1242
10. 10.What is the remainder when 132 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 1
11. 11.What is the remainder when 113 is divided by 7? (a) 0 (b) 2 (c) 6 (d) 1
12. 12.Which of these numbers is divisible by 9? (a) 3542 (b) 3528 (c) 3536 (d) 3524
13. 13.Which of these numbers is divisible by 4? (a) 1942 (b) 1947 (c) 1940 (d) 1939
14. 14.Which of these numbers is divisible by 3? (a) 2587 (b) 2593 (c) 2589 (d) 2591
15. 15.Which of these numbers is divisible by 9? (a) 6285 (b) 6265 (c) 6280 (d) 6273
16. 16.Which of these numbers is divisible by 6? (a) 4290 (b) 4295 (c) 4297 (d) 4288
17. 17.Which of these numbers is divisible by 6? (a) 3596 (b) 3600 (c) 3607 (d) 3603
18. 18.What is the units digit of 9^29? (a) 9 (b) 8 (c) 7 (d) 1
19. 19.What is the remainder when 68 is divided by 3? (a) 0 (b) 2 (c) 1
20. 20.What is the units digit of 4^55? (a) 6 (b) 3 (c) 4 (d) 5
21. 21.What is the smallest number you must add to 267 to make it divisible by 7? (a) 4 (b) 1 (c) 5 (d) 6
22. 22.Which of these numbers is divisible by 11? (a) 984 (b) 999 (c) 990 (d) 1003
23. 23.What is the units digit of 8^34? (a) 2 (b) 8 (c) 6 (d) 4
24. 24.Which of these numbers is divisible by 4? (a) 3239 (b) 3235 (c) 3236 (d) 3243
25. 25.What is the remainder when 371 is divided by 6? (a) 5 (b) 4 (c) 1 (d) 3
26. 26.What is the remainder when 383 is divided by 9? (a) 6 (b) 5 (c) 7 (d) 4
27. 27.What is the remainder when 177 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
28. 28.Which of these numbers is divisible by 8? (a) 1096 (b) 1089 (c) 1107 (d) 1097
29. 29.What is the remainder when 213 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
30. 30.What is the remainder when 650 is divided by 9? (a) 3 (b) 1 (c) 7 (d) 2
31. 31.Which of these numbers is divisible by 3? (a) 1388 (b) 1384 (c) 1381 (d) 1383
32. 32.What is the remainder when 679 is divided by 10? (a) 1 (b) 9 (c) 7 (d) 8
33. 33.What is the smallest number you must add to 290 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
34. 34.What is the units digit of 3^37? (a) 9 (b) 1 (c) 3 (d) 7
35. 35.Which of these numbers is divisible by 9? (a) 859 (b) 879 (c) 867 (d) 864
36. 36.What is the units digit of 9^54? (a) 0 (b) 2 (c) 1 (d) 9
37. 37.What is the remainder when 627 is divided by 12? (a) 9 (b) 2 (c) 4 (d) 3
38. 38.Which of these numbers is divisible by 8? (a) 5179 (b) 5188 (c) 5169 (d) 5176
39. 39.What is the smallest number you must add to 409 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 4
40. 40.What is the units digit of 2^10? (a) 2 (b) 6 (c) 4 (d) 8

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