Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 986 · 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 4? (a) 1289 (b) 1282 (c) 1284 (d) 1285
2. 2.What is the units digit of 3^27? (a) 7 (b) 9 (c) 3 (d) 1
3. 3.What is the remainder when 250 is divided by 9? (a) 2 (b) 7 (c) 8 (d) 6
4. 4.Which of these numbers is divisible by 3? (a) 2252 (b) 2247 (c) 2245 (d) 2248
5. 5.What is the remainder when 171 is divided by 5? (a) 2 (b) 1 (c) 0 (d) 4
6. 6.Which of these numbers is divisible by 4? (a) 916 (b) 913 (c) 921 (d) 919
7. 7.What is the smallest number you must add to 108 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 3
8. 8.What is the remainder when 264 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 2
9. 9.What is the smallest number you must add to 475 to make it divisible by 7? (a) 6 (b) 3 (c) 1 (d) 2
10. 10.Which of these numbers is divisible by 8? (a) 3517 (b) 3509 (c) 3526 (d) 3512
11. 11.What is the smallest number you must add to 454 to make it divisible by 10? (a) 5 (b) 4 (c) 6 (d) 7
12. 12.What is the remainder when 169 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
13. 13.What is the remainder when 160 is divided by 3? (a) 0 (b) 1 (c) 2
14. 14.What is the smallest number you must add to 284 to make it divisible by 10? (a) 7 (b) 5 (c) 6 (d) 4
15. 15.What is the units digit of 2^49? (a) 8 (b) 4 (c) 6 (d) 2
16. 16.What is the remainder when 179 is divided by 11? (a) 3 (b) 4 (c) 8 (d) 2
17. 17.Which of these numbers is divisible by 3? (a) 513 (b) 514 (c) 512 (d) 517
18. 18.What is the remainder when 930 is divided by 12? (a) 8 (b) 7 (c) 5 (d) 6
19. 19.Which of these numbers is divisible by 8? (a) 3167 (b) 3160 (c) 3173 (d) 3157
20. 20.What is the smallest number you must add to 154 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
21. 21.What is the units digit of 4^22? (a) 6 (b) 5 (c) 7 (d) 4
22. 22.What is the smallest number you must add to 405 to make it divisible by 11? (a) 9 (b) 2 (c) 3 (d) 1
23. 23.What is the units digit of 3^21? (a) 9 (b) 7 (c) 3 (d) 1
24. 24.Which of these numbers is divisible by 3? (a) 200 (b) 198 (c) 202 (d) 197
25. 25.Which of these numbers is divisible by 4? (a) 3360 (b) 3365 (c) 3359 (d) 3363
26. 26.Which of these numbers is divisible by 11? (a) 8370 (b) 8376 (c) 8383 (d) 8371
27. 27.What is the smallest number you must add to 254 to make it divisible by 8? (a) 3 (b) 6 (c) 1 (d) 2
28. 28.What is the units digit of 7^51? (a) 9 (b) 7 (c) 1 (d) 3
29. 29.What is the remainder when 146 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 1
30. 30.Which of these numbers is divisible by 9? (a) 1341 (b) 1351 (c) 1349 (d) 1335
31. 31.What is the smallest number you must add to 399 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
32. 32.What is the remainder when 316 is divided by 7? (a) 6 (b) 2 (c) 1 (d) 0
33. 33.What is the smallest number you must add to 230 to make it divisible by 4? (a) 2 (b) 3 (c) 1
34. 34.What is the units digit of 3^19? (a) 3 (b) 1 (c) 9 (d) 7
35. 35.What is the units digit of 7^43? (a) 3 (b) 7 (c) 1 (d) 9
36. 36.Which of these numbers is divisible by 8? (a) 1890 (b) 1873 (c) 1887 (d) 1880
37. 37.Which of these numbers is divisible by 4? (a) 773 (b) 777 (c) 772 (d) 771
38. 38.Which of these numbers is divisible by 9? (a) 3637 (b) 3624 (c) 3627 (d) 3629
39. 39.What is the remainder when 668 is divided by 9? (a) 3 (b) 1 (c) 7 (d) 2
40. 40.What is the smallest number you must add to 326 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1

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