Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 347 · 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 3? (a) 192 (b) 191 (c) 193 (d) 196
2. 2.Which of these numbers is divisible by 4? (a) 576 (b) 573 (c) 577 (d) 583
3. 3.What is the units digit of 3^12? (a) 3 (b) 1 (c) 7 (d) 9
4. 4.Which of these numbers is divisible by 4? (a) 1028 (b) 1030 (c) 1034 (d) 1026
5. 5.What is the smallest number you must add to 199 to make it divisible by 11? (a) 9 (b) 1 (c) 8 (d) 10
6. 6.What is the remainder when 175 is divided by 3? (a) 0 (b) 1 (c) 2
7. 7.What is the remainder when 405 is divided by 8? (a) 4 (b) 5 (c) 6 (d) 3
8. 8.Which of these numbers is divisible by 6? (a) 3152 (b) 3141 (c) 3144 (d) 3148
9. 9.Which of these numbers is divisible by 6? (a) 2691 (b) 2688 (c) 2686 (d) 2696
10. 10.What is the remainder when 647 is divided by 11? (a) 2 (b) 10 (c) 9 (d) 8
11. 11.What is the remainder when 209 is divided by 3? (a) 0 (b) 2 (c) 1
12. 12.Which of these numbers is divisible by 6? (a) 1308 (b) 1306 (c) 1313 (d) 1317
13. 13.What is the remainder when 229 is divided by 9? (a) 6 (b) 3 (c) 5 (d) 4
14. 14.What is the remainder when 556 is divided by 11? (a) 6 (b) 8 (c) 5 (d) 7
15. 15.What is the remainder when 205 is divided by 8? (a) 5 (b) 3 (c) 4 (d) 6
16. 16.What is the smallest number you must add to 378 to make it divisible by 11? (a) 4 (b) 7 (c) 8 (d) 6
17. 17.Which of these numbers is divisible by 3? (a) 2228 (b) 2224 (c) 2226 (d) 2231
18. 18.What is the smallest number you must add to 103 to make it divisible by 3? (a) 2 (b) 1
19. 19.What is the remainder when 298 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 5
20. 20.Which of these numbers is divisible by 8? (a) 4735 (b) 4720 (c) 4721 (d) 4719
21. 21.Which of these numbers is divisible by 9? (a) 2320 (b) 2304 (c) 2307 (d) 2299
22. 22.What is the units digit of 8^25? (a) 6 (b) 4 (c) 2 (d) 8
23. 23.What is the units digit of 9^24? (a) 0 (b) 1 (c) 2 (d) 9
24. 24.What is the remainder when 571 is divided by 10? (a) 9 (b) 0 (c) 1 (d) 2
25. 25.What is the units digit of 8^33? (a) 4 (b) 8 (c) 6 (d) 2
26. 26.Which of these numbers is divisible by 8? (a) 6233 (b) 6224 (c) 6225 (d) 6220
27. 27.Which of these numbers is divisible by 6? (a) 4620 (b) 4621 (c) 4616 (d) 4629
28. 28.What is the units digit of 3^46? (a) 9 (b) 1 (c) 3 (d) 7
29. 29.What is the remainder when 649 is divided by 12? (a) 1 (b) 11 (c) 2 (d) 0
30. 30.What is the remainder when 430 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
31. 31.What is the smallest number you must add to 142 to make it divisible by 8? (a) 1 (b) 6 (c) 2 (d) 3
32. 32.What is the remainder when 214 is divided by 3? (a) 2 (b) 0 (c) 1
33. 33.What is the units digit of 3^42? (a) 9 (b) 3 (c) 1 (d) 7
34. 34.What is the units digit of 3^55? (a) 1 (b) 7 (c) 9 (d) 3
35. 35.What is the units digit of 2^44? (a) 2 (b) 6 (c) 8 (d) 4
36. 36.Which of these numbers is divisible by 6? (a) 2863 (b) 2870 (c) 2859 (d) 2862
37. 37.Which of these numbers is divisible by 8? (a) 392 (b) 388 (c) 403 (d) 393
38. 38.What is the remainder when 206 is divided by 12? (a) 10 (b) 1 (c) 3 (d) 2
39. 39.Which of these numbers is divisible by 3? (a) 1390 (b) 1387 (c) 1393 (d) 1389
40. 40.What is the units digit of 2^43? (a) 2 (b) 4 (c) 8 (d) 6

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