Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 727 · 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 units digit of 9^40? (a) 2 (b) 1 (c) 0 (d) 9
2. 2.What is the smallest number you must add to 369 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
3. 3.What is the remainder when 431 is divided by 10? (a) 9 (b) 1 (c) 2 (d) 0
4. 4.What is the remainder when 297 is divided by 7? (a) 3 (b) 5 (c) 2 (d) 4
5. 5.What is the units digit of 2^57? (a) 4 (b) 8 (c) 6 (d) 2
6. 6.What is the units digit of 9^13? (a) 1 (b) 8 (c) 9 (d) 7
7. 7.What is the units digit of 4^56? (a) 7 (b) 4 (c) 6 (d) 5
8. 8.What is the smallest number you must add to 303 to make it divisible by 4? (a) 2 (b) 1 (c) 3
9. 9.What is the units digit of 7^11? (a) 7 (b) 9 (c) 1 (d) 3
10. 10.What is the units digit of 8^57? (a) 8 (b) 2 (c) 4 (d) 6
11. 11.Which of these numbers is divisible by 9? (a) 7710 (b) 7717 (c) 7713 (d) 7730
12. 12.What is the remainder when 854 is divided by 12? (a) 3 (b) 1 (c) 2 (d) 10
13. 13.What is the units digit of 3^44? (a) 7 (b) 1 (c) 9 (d) 3
14. 14.What is the smallest number you must add to 202 to make it divisible by 4? (a) 2 (b) 3 (c) 1
15. 15.Which of these numbers is divisible by 6? (a) 5004 (b) 5002 (c) 5008 (d) 5011
16. 16.Which of these numbers is divisible by 11? (a) 531 (b) 540 (c) 527 (d) 528
17. 17.What is the smallest number you must add to 177 to make it divisible by 4? (a) 1 (b) 2 (c) 3
18. 18.What is the remainder when 258 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
19. 19.What is the smallest number you must add to 261 to make it divisible by 12? (a) 9 (b) 4 (c) 2 (d) 3
20. 20.What is the remainder when 681 is divided by 9? (a) 5 (b) 7 (c) 6 (d) 3
21. 21.What is the smallest number you must add to 740 to make it divisible by 11? (a) 7 (b) 8 (c) 3 (d) 9
22. 22.What is the remainder when 520 is divided by 11? (a) 3 (b) 8 (c) 2 (d) 4
23. 23.Which of these numbers is divisible by 11? (a) 7273 (b) 7260 (c) 7250 (d) 7261
24. 24.Which of these numbers is divisible by 3? (a) 2299 (b) 2294 (c) 2295 (d) 2297
25. 25.Which of these numbers is divisible by 8? (a) 2184 (b) 2177 (c) 2198 (d) 2187
26. 26.Which of these numbers is divisible by 4? (a) 354 (b) 362 (c) 356 (d) 358
27. 27.What is the smallest number you must add to 62 to make it divisible by 4? (a) 3 (b) 1 (c) 2
28. 28.What is the smallest number you must add to 188 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
29. 29.What is the remainder when 101 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
30. 30.What is the smallest number you must add to 469 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 4
31. 31.What is the remainder when 116 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 2
32. 32.What is the smallest number you must add to 89 to make it divisible by 7? (a) 5 (b) 1 (c) 2 (d) 3
33. 33.What is the remainder when 186 is divided by 9? (a) 6 (b) 3 (c) 7 (d) 5
34. 34.What is the smallest number you must add to 707 to make it divisible by 12? (a) 2 (b) 11 (c) 1 (d) 3
35. 35.Which of these numbers is divisible by 3? (a) 1501 (b) 1499 (c) 1497 (d) 1495
36. 36.What is the remainder when 162 is divided by 7? (a) 0 (b) 6 (c) 1 (d) 2
37. 37.What is the units digit of 9^48? (a) 1 (b) 9 (c) 0 (d) 2
38. 38.What is the units digit of 9^30? (a) 2 (b) 9 (c) 1 (d) 0
39. 39.What is the remainder when 640 is divided by 12? (a) 3 (b) 8 (c) 4 (d) 5
40. 40.What is the remainder when 452 is divided by 9? (a) 3 (b) 1 (c) 7 (d) 2

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