Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 178 · 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 smallest number you must add to 271 to make it divisible by 10? (a) 8 (b) 7 (c) 1 (d) 9
2. 2.What is the units digit of 7^34? (a) 1 (b) 3 (c) 7 (d) 9
3. 3.What is the units digit of 7^40? (a) 3 (b) 7 (c) 1 (d) 9
4. 4.What is the units digit of 8^51? (a) 8 (b) 4 (c) 2 (d) 6
5. 5.What is the remainder when 197 is divided by 6? (a) 1 (b) 5 (c) 3 (d) 4
6. 6.What is the smallest number you must add to 303 to make it divisible by 8? (a) 1 (b) 2 (c) 7 (d) 3
7. 7.Which of these numbers is divisible by 9? (a) 5287 (b) 5280 (c) 5273 (d) 5274
8. 8.Which of these numbers is divisible by 11? (a) 9839 (b) 9860 (c) 9850 (d) 9845
9. 9.What is the smallest number you must add to 270 to make it divisible by 12? (a) 8 (b) 5 (c) 7 (d) 6
10. 10.Which of these numbers is divisible by 11? (a) 864 (b) 858 (c) 848 (d) 870
11. 11.What is the remainder when 127 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
12. 12.What is the remainder when 140 is divided by 9? (a) 5 (b) 4 (c) 6 (d) 7
13. 13.What is the remainder when 141 is divided by 8? (a) 3 (b) 4 (c) 6 (d) 5
14. 14.What is the remainder when 747 is divided by 12? (a) 2 (b) 3 (c) 9 (d) 4
15. 15.What is the remainder when 148 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 2
16. 16.Which of these numbers is divisible by 8? (a) 1140 (b) 1122 (c) 1129 (d) 1128
17. 17.What is the remainder when 517 is divided by 9? (a) 6 (b) 5 (c) 3 (d) 4
18. 18.Which of these numbers is divisible by 8? (a) 3527 (b) 3509 (c) 3512 (d) 3513
19. 19.What is the remainder when 227 is divided by 3? (a) 1 (b) 0 (c) 2
20. 20.What is the remainder when 446 is divided by 12? (a) 3 (b) 10 (c) 1 (d) 2
21. 21.Which of these numbers is divisible by 8? (a) 4610 (b) 4607 (c) 4600 (d) 4596
22. 22.What is the remainder when 257 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
23. 23.What is the remainder when 131 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
24. 24.Which of these numbers is divisible by 3? (a) 2336 (b) 2337 (c) 2342 (d) 2338
25. 25.What is the remainder when 169 is divided by 11? (a) 5 (b) 4 (c) 3 (d) 7
26. 26.Which of these numbers is divisible by 8? (a) 7084 (b) 7073 (c) 7080 (d) 7095
27. 27.What is the smallest number you must add to 519 to make it divisible by 7? (a) 6 (b) 1 (c) 4 (d) 5
28. 28.What is the smallest number you must add to 574 to make it divisible by 9? (a) 7 (b) 3 (c) 2 (d) 1
29. 29.What is the units digit of 3^15? (a) 3 (b) 9 (c) 1 (d) 7
30. 30.What is the smallest number you must add to 375 to make it divisible by 8? (a) 7 (b) 1 (c) 2 (d) 3
31. 31.What is the units digit of 3^28? (a) 9 (b) 7 (c) 1 (d) 3
32. 32.What is the smallest number you must add to 397 to make it divisible by 11? (a) 9 (b) 10 (c) 8 (d) 1
33. 33.What is the remainder when 113 is divided by 6? (a) 1 (b) 5 (c) 3 (d) 4
34. 34.Which of these numbers is divisible by 8? (a) 5044 (b) 5054 (c) 5040 (d) 5033
35. 35.What is the remainder when 357 is divided by 10? (a) 6 (b) 7 (c) 3 (d) 8
36. 36.What is the units digit of 3^47? (a) 7 (b) 3 (c) 1 (d) 9
37. 37.What is the units digit of 2^52? (a) 2 (b) 4 (c) 6 (d) 8
38. 38.Which of these numbers is divisible by 11? (a) 2992 (b) 2997 (c) 2986 (d) 3011
39. 39.Which of these numbers is divisible by 8? (a) 2785 (b) 2779 (c) 2795 (d) 2784
40. 40.What is the units digit of 2^28? (a) 8 (b) 4 (c) 6 (d) 2

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