Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 406 · 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 7^22? (a) 1 (b) 9 (c) 7 (d) 3
2. 2.What is the remainder when 156 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
3. 3.Which of these numbers is divisible by 8? (a) 4805 (b) 4792 (c) 4789 (d) 4793
4. 4.What is the remainder when 864 is divided by 11? (a) 7 (b) 6 (c) 5 (d) 8
5. 5.What is the units digit of 7^36? (a) 7 (b) 1 (c) 3 (d) 9
6. 6.Which of these numbers is divisible by 8? (a) 2229 (b) 2216 (c) 2209 (d) 2217
7. 7.What is the smallest number you must add to 236 to make it divisible by 3? (a) 2 (b) 1
8. 8.Which of these numbers is divisible by 6? (a) 594 (b) 595 (c) 601 (d) 589
9. 9.What is the remainder when 359 is divided by 10? (a) 9 (b) 7 (c) 8 (d) 1
10. 10.What is the remainder when 390 is divided by 11? (a) 5 (b) 7 (c) 6 (d) 4
11. 11.What is the remainder when 467 is divided by 6? (a) 5 (b) 4 (c) 1 (d) 3
12. 12.What is the units digit of 7^40? (a) 7 (b) 3 (c) 9 (d) 1
13. 13.What is the smallest number you must add to 196 to make it divisible by 8? (a) 5 (b) 6 (c) 3 (d) 4
14. 14.Which of these numbers is divisible by 3? (a) 1261 (b) 1263 (c) 1264 (d) 1267
15. 15.What is the units digit of 2^36? (a) 8 (b) 6 (c) 4 (d) 2
16. 16.What is the units digit of 9^23? (a) 1 (b) 7 (c) 8 (d) 9
17. 17.What is the units digit of 4^25? (a) 3 (b) 6 (c) 4 (d) 5
18. 18.Which of these numbers is divisible by 3? (a) 1367 (b) 1364 (c) 1362 (d) 1361
19. 19.What is the smallest number you must add to 153 to make it divisible by 11? (a) 2 (b) 3 (c) 1 (d) 10
20. 20.What is the remainder when 213 is divided by 9? (a) 7 (b) 5 (c) 3 (d) 6
21. 21.What is the units digit of 3^23? (a) 7 (b) 3 (c) 9 (d) 1
22. 22.Which of these numbers is divisible by 4? (a) 2009 (b) 2008 (c) 2013 (d) 2006
23. 23.What is the units digit of 4^32? (a) 5 (b) 7 (c) 6 (d) 4
24. 24.Which of these numbers is divisible by 9? (a) 4509 (b) 4526 (c) 4506 (d) 4515
25. 25.Which of these numbers is divisible by 6? (a) 1567 (b) 1577 (c) 1572 (d) 1583
26. 26.What is the smallest number you must add to 524 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 6
27. 27.Which of these numbers is divisible by 3? (a) 2296 (b) 2300 (c) 2294 (d) 2295
28. 28.Which of these numbers is divisible by 11? (a) 7073 (b) 7075 (c) 7064 (d) 7085
29. 29.Which of these numbers is divisible by 4? (a) 2595 (b) 2587 (c) 2588 (d) 2590
30. 30.What is the remainder when 172 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
31. 31.What is the smallest number you must add to 274 to make it divisible by 4? (a) 3 (b) 1 (c) 2
32. 32.Which of these numbers is divisible by 4? (a) 902 (b) 897 (c) 907 (d) 900
33. 33.What is the remainder when 134 is divided by 3? (a) 0 (b) 2 (c) 1
34. 34.Which of these numbers is divisible by 4? (a) 2470 (b) 2466 (c) 2475 (d) 2468
35. 35.Which of these numbers is divisible by 4? (a) 1788 (b) 1787 (c) 1790 (d) 1794
36. 36.Which of these numbers is divisible by 11? (a) 9848 (b) 9827 (c) 9834 (d) 9837
37. 37.Which of these numbers is divisible by 11? (a) 6782 (b) 6787 (c) 6795 (d) 6806
38. 38.What is the units digit of 4^49? (a) 4 (b) 3 (c) 6 (d) 5
39. 39.What is the smallest number you must add to 337 to make it divisible by 7? (a) 6 (b) 1 (c) 4 (d) 5
40. 40.What is the remainder when 124 is divided by 3? (a) 0 (b) 1 (c) 2

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