Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 693 · 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 3^32? (a) 7 (b) 9 (c) 3 (d) 1
2. 2.What is the smallest number you must add to 668 to make it divisible by 12? (a) 5 (b) 3 (c) 8 (d) 4
3. 3.What is the remainder when 718 is divided by 9? (a) 8 (b) 7 (c) 6 (d) 2
4. 4.What is the units digit of 7^15? (a) 3 (b) 7 (c) 9 (d) 1
5. 5.What is the units digit of 3^42? (a) 9 (b) 7 (c) 1 (d) 3
6. 6.What is the remainder when 471 is divided by 10? (a) 2 (b) 1 (c) 0 (d) 9
7. 7.Which of these numbers is divisible by 9? (a) 3056 (b) 3036 (c) 3042 (d) 3049
8. 8.Which of these numbers is divisible by 11? (a) 2090 (b) 2105 (c) 2095 (d) 2081
9. 9.What is the smallest number you must add to 241 to make it divisible by 7? (a) 4 (b) 5 (c) 3 (d) 6
10. 10.Which of these numbers is divisible by 4? (a) 2078 (b) 2080 (c) 2086 (d) 2081
11. 11.What is the remainder when 307 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
12. 12.What is the smallest number you must add to 330 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 2
13. 13.Which of these numbers is divisible by 3? (a) 752 (b) 745 (c) 748 (d) 747
14. 14.What is the units digit of 9^27? (a) 8 (b) 1 (c) 7 (d) 9
15. 15.What is the units digit of 2^52? (a) 2 (b) 8 (c) 4 (d) 6
16. 16.Which of these numbers is divisible by 8? (a) 1413 (b) 1405 (c) 1400 (d) 1396
17. 17.Which of these numbers is divisible by 4? (a) 3477 (b) 3483 (c) 3480 (d) 3487
18. 18.Which of these numbers is divisible by 11? (a) 6030 (b) 6028 (c) 6041 (d) 6022
19. 19.What is the smallest number you must add to 582 to make it divisible by 8? (a) 6 (b) 3 (c) 2 (d) 1
20. 20.What is the units digit of 4^60? (a) 4 (b) 6 (c) 7 (d) 5
21. 21.Which of these numbers is divisible by 8? (a) 2240 (b) 2243 (c) 2251 (d) 2234
22. 22.Which of these numbers is divisible by 4? (a) 384 (b) 386 (c) 389 (d) 381
23. 23.What is the remainder when 251 is divided by 12? (a) 10 (b) 11 (c) 9 (d) 1
24. 24.What is the remainder when 449 is divided by 8? (a) 0 (b) 7 (c) 1 (d) 2
25. 25.Which of these numbers is divisible by 11? (a) 3036 (b) 3053 (c) 3031 (d) 3041
26. 26.What is the units digit of 2^25? (a) 8 (b) 6 (c) 4 (d) 2
27. 27.What is the remainder when 305 is divided by 10? (a) 6 (b) 5 (c) 7 (d) 4
28. 28.What is the smallest number you must add to 254 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
29. 29.What is the smallest number you must add to 89 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
30. 30.What is the remainder when 322 is divided by 10? (a) 1 (b) 3 (c) 2 (d) 8
31. 31.What is the remainder when 281 is divided by 11? (a) 7 (b) 8 (c) 5 (d) 6
32. 32.What is the smallest number you must add to 419 to make it divisible by 8? (a) 3 (b) 6 (c) 4 (d) 5
33. 33.Which of these numbers is divisible by 3? (a) 1979 (b) 1980 (c) 1981 (d) 1985
34. 34.Which of these numbers is divisible by 6? (a) 4050 (b) 4061 (c) 4049 (d) 4051
35. 35.What is the smallest number you must add to 42 to make it divisible by 4? (a) 3 (b) 1 (c) 2
36. 36.What is the remainder when 327 is divided by 12? (a) 2 (b) 4 (c) 9 (d) 3
37. 37.Which of these numbers is divisible by 3? (a) 268 (b) 267 (c) 271 (d) 265
38. 38.Which of these numbers is divisible by 6? (a) 2144 (b) 2140 (c) 2142 (d) 2149
39. 39.Which of these numbers is divisible by 9? (a) 1017 (b) 1011 (c) 1020 (d) 1027
40. 40.What is the remainder when 614 is divided by 8? (a) 6 (b) 7 (c) 2 (d) 5

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