Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 26 · 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.Which of these numbers is divisible by 6? (a) 859 (b) 865 (c) 858 (d) 853
2. 2.What is the units digit of 3^11? (a) 7 (b) 1 (c) 3 (d) 9
3. 3.What is the units digit of 9^36? (a) 9 (b) 2 (c) 1 (d) 0
4. 4.What is the units digit of 2^20? (a) 6 (b) 8 (c) 2 (d) 4
5. 5.What is the remainder when 318 is divided by 9? (a) 2 (b) 3 (c) 6 (d) 4
6. 6.Which of these numbers is divisible by 4? (a) 2751 (b) 2759 (c) 2752 (d) 2754
7. 7.What is the remainder when 194 is divided by 3? (a) 2 (b) 1 (c) 0
8. 8.What is the units digit of 9^16? (a) 2 (b) 1 (c) 0 (d) 9
9. 9.Which of these numbers is divisible by 8? (a) 3934 (b) 3928 (c) 3924 (d) 3937
10. 10.What is the units digit of 7^34? (a) 1 (b) 3 (c) 7 (d) 9
11. 11.What is the remainder when 299 is divided by 8? (a) 3 (b) 2 (c) 5 (d) 4
12. 12.Which of these numbers is divisible by 3? (a) 781 (b) 779 (c) 775 (d) 777
13. 13.What is the smallest number you must add to 380 to make it divisible by 9? (a) 7 (b) 2 (c) 6 (d) 8
14. 14.What is the smallest number you must add to 206 to make it divisible by 8? (a) 1 (b) 2 (c) 3 (d) 6
15. 15.Which of these numbers is divisible by 4? (a) 1686 (b) 1689 (c) 1684 (d) 1683
16. 16.Which of these numbers is divisible by 11? (a) 929 (b) 935 (c) 938 (d) 955
17. 17.What is the remainder when 865 is divided by 12? (a) 1 (b) 0 (c) 2 (d) 11
18. 18.What is the remainder when 309 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
19. 19.What is the units digit of 7^56? (a) 3 (b) 7 (c) 1 (d) 9
20. 20.What is the units digit of 9^28? (a) 0 (b) 9 (c) 1 (d) 2
21. 21.Which of these numbers is divisible by 4? (a) 170 (b) 167 (c) 168 (d) 173
22. 22.What is the remainder when 87 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
23. 23.What is the smallest number you must add to 311 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 2
24. 24.What is the smallest number you must add to 727 to make it divisible by 10? (a) 4 (b) 2 (c) 7 (d) 3
25. 25.What is the remainder when 315 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
26. 26.Which of these numbers is divisible by 6? (a) 3157 (b) 3166 (c) 3153 (d) 3156
27. 27.What is the smallest number you must add to 237 to make it divisible by 9? (a) 7 (b) 5 (c) 6 (d) 3
28. 28.What is the remainder when 393 is divided by 7? (a) 1 (b) 2 (c) 6 (d) 0
29. 29.What is the remainder when 463 is divided by 7? (a) 6 (b) 0 (c) 2 (d) 1
30. 30.What is the units digit of 7^31? (a) 3 (b) 9 (c) 1 (d) 7
31. 31.Which of these numbers is divisible by 8? (a) 3412 (b) 3400 (c) 3401 (d) 3399
32. 32.What is the units digit of 8^49? (a) 8 (b) 6 (c) 4 (d) 2
33. 33.What is the units digit of 3^32? (a) 9 (b) 1 (c) 7 (d) 3
34. 34.What is the smallest number you must add to 177 to make it divisible by 4? (a) 1 (b) 2 (c) 3
35. 35.What is the remainder when 336 is divided by 9? (a) 2 (b) 4 (c) 3 (d) 6
36. 36.What is the units digit of 2^24? (a) 4 (b) 8 (c) 2 (d) 6
37. 37.Which of these numbers is divisible by 6? (a) 4935 (b) 4926 (c) 4928 (d) 4923
38. 38.Which of these numbers is divisible by 4? (a) 1657 (b) 1655 (c) 1656 (d) 1661
39. 39.What is the smallest number you must add to 468 to make it divisible by 8? (a) 3 (b) 6 (c) 4 (d) 5
40. 40.What is the units digit of 4^10? (a) 7 (b) 4 (c) 6 (d) 5

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