Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 788 · 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 197 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 1
2. 2.Which of these numbers is divisible by 8? (a) 6583 (b) 6576 (c) 6590 (d) 6570
3. 3.Which of these numbers is divisible by 9? (a) 4570 (b) 4562 (c) 4563 (d) 4575
4. 4.What is the smallest number you must add to 309 to make it divisible by 4? (a) 1 (b) 2 (c) 3
5. 5.Which of these numbers is divisible by 8? (a) 7018 (b) 7008 (c) 7015 (d) 7006
6. 6.What is the remainder when 251 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
7. 7.What is the units digit of 8^25? (a) 6 (b) 8 (c) 2 (d) 4
8. 8.What is the remainder when 399 is divided by 12? (a) 3 (b) 9 (c) 2 (d) 4
9. 9.What is the units digit of 4^26? (a) 5 (b) 4 (c) 6 (d) 7
10. 10.What is the remainder when 363 is divided by 10? (a) 2 (b) 4 (c) 7 (d) 3
11. 11.What is the remainder when 357 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
12. 12.Which of these numbers is divisible by 4? (a) 2476 (b) 2478 (c) 2482 (d) 2475
13. 13.Which of these numbers is divisible by 4? (a) 970 (b) 966 (c) 968 (d) 974
14. 14.What is the smallest number you must add to 497 to make it divisible by 9? (a) 6 (b) 8 (c) 2 (d) 7
15. 15.Which of these numbers is divisible by 3? (a) 2037 (b) 2042 (c) 2036 (d) 2038
16. 16.What is the units digit of 2^52? (a) 4 (b) 8 (c) 2 (d) 6
17. 17.What is the units digit of 4^54? (a) 7 (b) 6 (c) 5 (d) 4
18. 18.What is the units digit of 9^11? (a) 9 (b) 1 (c) 7 (d) 8
19. 19.What is the remainder when 381 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
20. 20.Which of these numbers is divisible by 9? (a) 2023 (b) 2007 (c) 2010 (d) 2003
21. 21.What is the remainder when 381 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
22. 22.What is the units digit of 3^26? (a) 3 (b) 9 (c) 1 (d) 7
23. 23.Which of these numbers is divisible by 11? (a) 2739 (b) 2758 (c) 2735 (d) 2742
24. 24.What is the smallest number you must add to 738 to make it divisible by 11? (a) 9 (b) 10 (c) 8 (d) 1
25. 25.What is the units digit of 7^19? (a) 1 (b) 7 (c) 9 (d) 3
26. 26.What is the smallest number you must add to 100 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 2
27. 27.What is the smallest number you must add to 206 to make it divisible by 7? (a) 3 (b) 5 (c) 6 (d) 4
28. 28.Which of these numbers is divisible by 11? (a) 3679 (b) 3657 (c) 3663 (d) 3672
29. 29.What is the smallest number you must add to 402 to make it divisible by 11? (a) 4 (b) 6 (c) 5 (d) 7
30. 30.What is the smallest number you must add to 421 to make it divisible by 8? (a) 4 (b) 2 (c) 5 (d) 3
31. 31.What is the units digit of 7^11? (a) 9 (b) 7 (c) 1 (d) 3
32. 32.What is the remainder when 273 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
33. 33.Which of these numbers is divisible by 9? (a) 2213 (b) 2222 (c) 2202 (d) 2205
34. 34.What is the units digit of 4^14? (a) 6 (b) 5 (c) 7 (d) 4
35. 35.What is the remainder when 968 is divided by 12? (a) 8 (b) 9 (c) 7 (d) 4
36. 36.What is the smallest number you must add to 296 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
37. 37.What is the remainder when 140 is divided by 9? (a) 4 (b) 6 (c) 7 (d) 5
38. 38.Which of these numbers is divisible by 8? (a) 2651 (b) 2642 (c) 2634 (d) 2640
39. 39.What is the remainder when 311 is divided by 8? (a) 1 (b) 5 (c) 6 (d) 7
40. 40.Which of these numbers is divisible by 4? (a) 1673 (b) 1677 (c) 1672 (d) 1669

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