Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 256 · 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 2^26? (a) 6 (b) 8 (c) 2 (d) 4
2. 2.What is the remainder when 147 is divided by 12? (a) 2 (b) 3 (c) 4 (d) 9
3. 3.What is the remainder when 358 is divided by 9? (a) 2 (b) 7 (c) 6 (d) 8
4. 4.What is the units digit of 9^10? (a) 2 (b) 0 (c) 1 (d) 9
5. 5.Which of these numbers is divisible by 6? (a) 4227 (b) 4230 (c) 4233 (d) 4241
6. 6.What is the units digit of 3^23? (a) 1 (b) 9 (c) 7 (d) 3
7. 7.What is the remainder when 606 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
8. 8.What is the units digit of 3^45? (a) 9 (b) 1 (c) 3 (d) 7
9. 9.What is the smallest number you must add to 103 to make it divisible by 3? (a) 2 (b) 1
10. 10.Which of these numbers is divisible by 3? (a) 2297 (b) 2292 (c) 2294 (d) 2291
11. 11.What is the units digit of 7^33? (a) 9 (b) 3 (c) 1 (d) 7
12. 12.What is the remainder when 206 is divided by 3? (a) 2 (b) 0 (c) 1
13. 13.What is the remainder when 141 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
14. 14.Which of these numbers is divisible by 6? (a) 4605 (b) 4617 (c) 4608 (d) 4613
15. 15.Which of these numbers is divisible by 4? (a) 1577 (b) 1581 (c) 1580 (d) 1587
16. 16.What is the units digit of 4^37? (a) 6 (b) 3 (c) 5 (d) 4
17. 17.What is the remainder when 152 is divided by 10? (a) 2 (b) 8 (c) 3 (d) 1
18. 18.What is the units digit of 8^49? (a) 6 (b) 2 (c) 8 (d) 4
19. 19.Which of these numbers is divisible by 11? (a) 9462 (b) 9460 (c) 9459 (d) 9475
20. 20.What is the units digit of 7^50? (a) 7 (b) 3 (c) 1 (d) 9
21. 21.What is the remainder when 158 is divided by 3? (a) 1 (b) 0 (c) 2
22. 22.What is the smallest number you must add to 54 to make it divisible by 4? (a) 1 (b) 3 (c) 2
23. 23.Which of these numbers is divisible by 4? (a) 1885 (b) 1882 (c) 1884 (d) 1891
24. 24.What is the smallest number you must add to 529 to make it divisible by 8? (a) 5 (b) 6 (c) 7 (d) 1
25. 25.What is the units digit of 8^56? (a) 8 (b) 2 (c) 4 (d) 6
26. 26.Which of these numbers is divisible by 9? (a) 5323 (b) 5310 (c) 5309 (d) 5318
27. 27.What is the units digit of 9^45? (a) 9 (b) 8 (c) 1 (d) 7
28. 28.What is the units digit of 2^46? (a) 8 (b) 6 (c) 4 (d) 2
29. 29.What is the units digit of 3^17? (a) 1 (b) 7 (c) 9 (d) 3
30. 30.What is the remainder when 194 is divided by 3? (a) 1 (b) 0 (c) 2
31. 31.What is the units digit of 2^48? (a) 2 (b) 8 (c) 6 (d) 4
32. 32.What is the smallest number you must add to 162 to make it divisible by 7? (a) 6 (b) 4 (c) 1 (d) 5
33. 33.What is the remainder when 237 is divided by 12? (a) 10 (b) 8 (c) 3 (d) 9
34. 34.What is the remainder when 894 is divided by 12? (a) 8 (b) 7 (c) 6 (d) 5
35. 35.Which of these numbers is divisible by 8? (a) 2952 (b) 2950 (c) 2953 (d) 2964
36. 36.Which of these numbers is divisible by 3? (a) 313 (b) 308 (c) 311 (d) 309
37. 37.Which of these numbers is divisible by 6? (a) 3449 (b) 3434 (c) 3441 (d) 3438
38. 38.Which of these numbers is divisible by 9? (a) 804 (b) 792 (c) 800 (d) 784
39. 39.Which of these numbers is divisible by 8? (a) 6194 (b) 6184 (c) 6183 (d) 6190
40. 40.What is the units digit of 2^42? (a) 6 (b) 4 (c) 2 (d) 8

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