Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 289 · 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 smallest number you must add to 94 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
2. 2.What is the remainder when 263 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
3. 3.What is the smallest number you must add to 644 to make it divisible by 9? (a) 5 (b) 6 (c) 3 (d) 4
4. 4.Which of these numbers is divisible by 4? (a) 1437 (b) 1430 (c) 1433 (d) 1432
5. 5.What is the units digit of 7^60? (a) 1 (b) 3 (c) 9 (d) 7
6. 6.What is the smallest number you must add to 242 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
7. 7.Which of these numbers is divisible by 4? (a) 1863 (b) 1856 (c) 1859 (d) 1853
8. 8.What is the units digit of 9^11? (a) 7 (b) 9 (c) 8 (d) 1
9. 9.What is the units digit of 4^30? (a) 7 (b) 4 (c) 6 (d) 5
10. 10.What is the smallest number you must add to 339 to make it divisible by 9? (a) 2 (b) 4 (c) 3 (d) 6
11. 11.What is the smallest number you must add to 423 to make it divisible by 12? (a) 8 (b) 3 (c) 10 (d) 9
12. 12.What is the smallest number you must add to 136 to make it divisible by 3? (a) 1 (b) 2
13. 13.What is the remainder when 522 is divided by 7? (a) 6 (b) 3 (c) 5 (d) 4
14. 14.What is the units digit of 8^55? (a) 8 (b) 6 (c) 2 (d) 4
15. 15.What is the smallest number you must add to 102 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
16. 16.Which of these numbers is divisible by 3? (a) 2009 (b) 2011 (c) 2006 (d) 2007
17. 17.Which of these numbers is divisible by 9? (a) 4714 (b) 4706 (c) 4721 (d) 4707
18. 18.What is the remainder when 199 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
19. 19.What is the remainder when 236 is divided by 12? (a) 9 (b) 4 (c) 7 (d) 8
20. 20.Which of these numbers is divisible by 3? (a) 1761 (b) 1766 (c) 1763 (d) 1759
21. 21.What is the remainder when 158 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 1
22. 22.Which of these numbers is divisible by 3? (a) 1395 (b) 1397 (c) 1393 (d) 1400
23. 23.Which of these numbers is divisible by 6? (a) 2435 (b) 2441 (c) 2430 (d) 2429
24. 24.Which of these numbers is divisible by 11? (a) 6471 (b) 6454 (c) 6457 (d) 6460
25. 25.What is the units digit of 2^57? (a) 6 (b) 4 (c) 2 (d) 8
26. 26.What is the remainder when 421 is divided by 11? (a) 2 (b) 8 (c) 3 (d) 4
27. 27.What is the units digit of 2^41? (a) 4 (b) 2 (c) 6 (d) 8
28. 28.What is the remainder when 237 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
29. 29.Which of these numbers is divisible by 9? (a) 1282 (b) 1263 (c) 1272 (d) 1269
30. 30.Which of these numbers is divisible by 9? (a) 7977 (b) 7974 (c) 7967 (d) 7985
31. 31.What is the smallest number you must add to 118 to make it divisible by 7? (a) 6 (b) 3 (c) 2 (d) 1
32. 32.What is the units digit of 3^56? (a) 3 (b) 9 (c) 1 (d) 7
33. 33.Which of these numbers is divisible by 8? (a) 2182 (b) 2194 (c) 2190 (d) 2184
34. 34.Which of these numbers is divisible by 11? (a) 1528 (b) 1532 (c) 1529 (d) 1547
35. 35.What is the remainder when 220 is divided by 3? (a) 1 (b) 0 (c) 2
36. 36.What is the units digit of 8^50? (a) 6 (b) 2 (c) 4 (d) 8
37. 37.What is the remainder when 198 is divided by 7? (a) 3 (b) 5 (c) 1 (d) 2
38. 38.What is the remainder when 139 is divided by 6? (a) 0 (b) 1 (c) 2 (d) 5
39. 39.What is the smallest number you must add to 486 to make it divisible by 11? (a) 9 (b) 2 (c) 10 (d) 8
40. 40.What is the smallest number you must add to 405 to make it divisible by 12? (a) 4 (b) 9 (c) 2 (d) 3

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