Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 481 · 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.Which of these numbers is divisible by 8? (a) 3950 (b) 3954 (c) 3942 (d) 3944
2. 2.Which of these numbers is divisible by 3? (a) 2266 (b) 2265 (c) 2269 (d) 2264
3. 3.Which of these numbers is divisible by 4? (a) 1213 (b) 1207 (c) 1208 (d) 1211
4. 4.What is the smallest number you must add to 321 to make it divisible by 8? (a) 6 (b) 1 (c) 5 (d) 7
5. 5.Which of these numbers is divisible by 9? (a) 6383 (b) 6402 (c) 6390 (d) 6397
6. 6.What is the remainder when 293 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
7. 7.Which of these numbers is divisible by 9? (a) 1982 (b) 1997 (c) 1980 (d) 1976
8. 8.What is the remainder when 201 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
9. 9.What is the units digit of 3^33? (a) 1 (b) 3 (c) 7 (d) 9
10. 10.Which of these numbers is divisible by 8? (a) 6117 (b) 6121 (c) 6120 (d) 6135
11. 11.What is the smallest number you must add to 316 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
12. 12.What is the smallest number you must add to 85 to make it divisible by 4? (a) 1 (b) 2 (c) 3
13. 13.Which of these numbers is divisible by 6? (a) 1866 (b) 1876 (c) 1871 (d) 1865
14. 14.Which of these numbers is divisible by 9? (a) 7879 (b) 7892 (c) 7884 (d) 7899
15. 15.What is the units digit of 8^22? (a) 8 (b) 6 (c) 2 (d) 4
16. 16.Which of these numbers is divisible by 8? (a) 1694 (b) 1681 (c) 1697 (d) 1688
17. 17.What is the units digit of 2^18? (a) 4 (b) 6 (c) 8 (d) 2
18. 18.What is the smallest number you must add to 51 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
19. 19.What is the units digit of 3^10? (a) 1 (b) 7 (c) 3 (d) 9
20. 20.What is the units digit of 8^23? (a) 6 (b) 4 (c) 2 (d) 8
21. 21.What is the units digit of 7^23? (a) 1 (b) 9 (c) 7 (d) 3
22. 22.What is the smallest number you must add to 73 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
23. 23.Which of these numbers is divisible by 8? (a) 6433 (b) 6427 (c) 6419 (d) 6424
24. 24.What is the smallest number you must add to 102 to make it divisible by 4? (a) 3 (b) 1 (c) 2
25. 25.What is the smallest number you must add to 185 to make it divisible by 10? (a) 5 (b) 7 (c) 4 (d) 6
26. 26.What is the units digit of 4^47? (a) 3 (b) 4 (c) 5 (d) 6
27. 27.Which of these numbers is divisible by 6? (a) 2479 (b) 2470 (c) 2472 (d) 2474
28. 28.What is the units digit of 8^54? (a) 6 (b) 8 (c) 4 (d) 2
29. 29.Which of these numbers is divisible by 4? (a) 603 (b) 607 (c) 610 (d) 604
30. 30.What is the units digit of 9^37? (a) 9 (b) 8 (c) 7 (d) 1
31. 31.Which of these numbers is divisible by 3? (a) 1785 (b) 1787 (c) 1790 (d) 1784
32. 32.Which of these numbers is divisible by 6? (a) 4074 (b) 4084 (c) 4071 (d) 4077
33. 33.Which of these numbers is divisible by 11? (a) 5950 (b) 5959 (c) 5951 (d) 5970
34. 34.What is the smallest number you must add to 92 to make it divisible by 3? (a) 1 (b) 2
35. 35.What is the units digit of 4^42? (a) 4 (b) 7 (c) 5 (d) 6
36. 36.What is the remainder when 191 is divided by 7? (a) 1 (b) 2 (c) 5 (d) 3
37. 37.What is the remainder when 375 is divided by 12? (a) 3 (b) 9 (c) 2 (d) 4
38. 38.Which of these numbers is divisible by 8? (a) 4460 (b) 4466 (c) 4475 (d) 4464
39. 39.What is the units digit of 2^59? (a) 6 (b) 8 (c) 2 (d) 4
40. 40.What is the units digit of 4^59? (a) 4 (b) 6 (c) 5 (d) 3

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