Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 977 · 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 187 is divided by 9? (a) 6 (b) 2 (c) 8 (d) 7
2. 2.Which of these numbers is divisible by 4? (a) 408 (b) 407 (c) 413 (d) 409
3. 3.What is the units digit of 3^45? (a) 7 (b) 1 (c) 3 (d) 9
4. 4.What is the remainder when 298 is divided by 12? (a) 9 (b) 11 (c) 2 (d) 10
5. 5.Which of these numbers is divisible by 6? (a) 4497 (b) 4500 (c) 4511 (d) 4501
6. 6.What is the smallest number you must add to 379 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 4
7. 7.What is the smallest number you must add to 542 to make it divisible by 7? (a) 4 (b) 5 (c) 3 (d) 6
8. 8.What is the units digit of 2^40? (a) 4 (b) 2 (c) 6 (d) 8
9. 9.What is the units digit of 3^48? (a) 3 (b) 1 (c) 7 (d) 9
10. 10.What is the units digit of 2^27? (a) 2 (b) 4 (c) 8 (d) 6
11. 11.What is the remainder when 297 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
12. 12.What is the units digit of 8^19? (a) 6 (b) 4 (c) 2 (d) 8
13. 13.What is the units digit of 3^36? (a) 9 (b) 7 (c) 1 (d) 3
14. 14.What is the units digit of 8^55? (a) 6 (b) 8 (c) 2 (d) 4
15. 15.Which of these numbers is divisible by 6? (a) 3138 (b) 3140 (c) 3135 (d) 3146
16. 16.What is the remainder when 398 is divided by 6? (a) 1 (b) 2 (c) 3 (d) 4
17. 17.What is the units digit of 9^33? (a) 8 (b) 9 (c) 1 (d) 7
18. 18.Which of these numbers is divisible by 8? (a) 650 (b) 665 (c) 656 (d) 661
19. 19.Which of these numbers is divisible by 11? (a) 1849 (b) 1869 (c) 1839 (d) 1848
20. 20.What is the units digit of 8^33? (a) 2 (b) 6 (c) 4 (d) 8
21. 21.Which of these numbers is divisible by 4? (a) 2235 (b) 2237 (c) 2229 (d) 2232
22. 22.Which of these numbers is divisible by 4? (a) 584 (b) 589 (c) 585 (d) 582
23. 23.Which of these numbers is divisible by 11? (a) 1910 (b) 1903 (c) 1920 (d) 1893
24. 24.What is the remainder when 150 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 4
25. 25.What is the remainder when 159 is divided by 9? (a) 7 (b) 5 (c) 6 (d) 3
26. 26.Which of these numbers is divisible by 6? (a) 2253 (b) 2256 (c) 2257 (d) 2266
27. 27.What is the units digit of 9^48? (a) 9 (b) 1 (c) 2 (d) 0
28. 28.What is the remainder when 267 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
29. 29.What is the units digit of 7^32? (a) 3 (b) 7 (c) 9 (d) 1
30. 30.Which of these numbers is divisible by 9? (a) 4860 (b) 4856 (c) 4875 (d) 4867
31. 31.What is the smallest number you must add to 153 to make it divisible by 4? (a) 1 (b) 2 (c) 3
32. 32.What is the remainder when 282 is divided by 10? (a) 1 (b) 3 (c) 2 (d) 8
33. 33.Which of these numbers is divisible by 11? (a) 8983 (b) 8987 (c) 8988 (d) 9008
34. 34.Which of these numbers is divisible by 3? (a) 2455 (b) 2457 (c) 2461 (d) 2459
35. 35.What is the remainder when 713 is divided by 10? (a) 2 (b) 4 (c) 7 (d) 3
36. 36.What is the remainder when 705 is divided by 9? (a) 2 (b) 4 (c) 6 (d) 3
37. 37.Which of these numbers is divisible by 6? (a) 1674 (b) 1685 (c) 1677 (d) 1670
38. 38.Which of these numbers is divisible by 11? (a) 4718 (b) 4724 (c) 4704 (d) 4708
39. 39.What is the remainder when 154 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
40. 40.What is the units digit of 7^21? (a) 9 (b) 7 (c) 3 (d) 1

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