Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 805 · 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 remainder when 92 is divided by 8? (a) 6 (b) 4 (c) 5 (d) 3
2. 2.Which of these numbers is divisible by 11? (a) 8330 (b) 8346 (c) 8327 (d) 8326
3. 3.What is the remainder when 229 is divided by 3? (a) 2 (b) 0 (c) 1
4. 4.What is the units digit of 4^50? (a) 4 (b) 7 (c) 5 (d) 6
5. 5.What is the smallest number you must add to 412 to make it divisible by 9? (a) 1 (b) 2 (c) 7 (d) 3
6. 6.What is the smallest number you must add to 43 to make it divisible by 3? (a) 2 (b) 1
7. 7.What is the remainder when 398 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
8. 8.Which of these numbers is divisible by 6? (a) 4918 (b) 4921 (c) 4930 (d) 4920
9. 9.What is the units digit of 8^36? (a) 2 (b) 4 (c) 6 (d) 8
10. 10.What is the smallest number you must add to 98 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the remainder when 319 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
12. 12.Which of these numbers is divisible by 4? (a) 1293 (b) 1292 (c) 1291 (d) 1299
13. 13.What is the units digit of 7^17? (a) 7 (b) 3 (c) 9 (d) 1
14. 14.What is the remainder when 213 is divided by 9? (a) 7 (b) 3 (c) 6 (d) 5
15. 15.What is the remainder when 246 is divided by 5? (a) 1 (b) 0 (c) 2 (d) 4
16. 16.What is the smallest number you must add to 197 to make it divisible by 3? (a) 1 (b) 2
17. 17.What is the smallest number you must add to 125 to make it divisible by 3? (a) 2 (b) 1
18. 18.What is the smallest number you must add to 455 to make it divisible by 9? (a) 3 (b) 4 (c) 6 (d) 5
19. 19.Which of these numbers is divisible by 3? (a) 2661 (b) 2662 (c) 2659 (d) 2666
20. 20.What is the remainder when 367 is divided by 12? (a) 5 (b) 6 (c) 7 (d) 8
21. 21.What is the remainder when 264 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 2
22. 22.What is the remainder when 258 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
23. 23.What is the remainder when 142 is divided by 3? (a) 2 (b) 1 (c) 0
24. 24.What is the smallest number you must add to 183 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 7
25. 25.Which of these numbers is divisible by 11? (a) 2459 (b) 2478 (c) 2464 (d) 2466
26. 26.What is the smallest number you must add to 388 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
27. 27.Which of these numbers is divisible by 8? (a) 6012 (b) 5996 (c) 6000 (d) 6005
28. 28.Which of these numbers is divisible by 8? (a) 6508 (b) 6493 (c) 6496 (d) 6502
29. 29.What is the units digit of 3^29? (a) 9 (b) 7 (c) 3 (d) 1
30. 30.Which of these numbers is divisible by 3? (a) 319 (b) 317 (c) 313 (d) 315
31. 31.Which of these numbers is divisible by 3? (a) 485 (b) 487 (c) 486 (d) 490
32. 32.What is the remainder when 154 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
33. 33.Which of these numbers is divisible by 3? (a) 1248 (b) 1246 (c) 1253 (d) 1249
34. 34.What is the remainder when 336 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
35. 35.What is the smallest number you must add to 253 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
36. 36.What is the smallest number you must add to 184 to make it divisible by 12? (a) 9 (b) 8 (c) 4 (d) 7
37. 37.What is the units digit of 7^22? (a) 3 (b) 7 (c) 9 (d) 1
38. 38.What is the remainder when 169 is divided by 3? (a) 2 (b) 0 (c) 1
39. 39.Which of these numbers is divisible by 4? (a) 1850 (b) 1845 (c) 1841 (d) 1844
40. 40.What is the remainder when 328 is divided by 6? (a) 2 (b) 5 (c) 3 (d) 4

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