Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 258 · 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 7^20? (a) 1 (b) 9 (c) 3 (d) 7
2. 2.What is the units digit of 7^28? (a) 7 (b) 1 (c) 3 (d) 9
3. 3.Which of these numbers is divisible by 9? (a) 7817 (b) 7823 (c) 7804 (d) 7812
4. 4.What is the units digit of 2^13? (a) 4 (b) 8 (c) 6 (d) 2
5. 5.Which of these numbers is divisible by 11? (a) 1730 (b) 1716 (c) 1712 (d) 1726
6. 6.Which of these numbers is divisible by 6? (a) 3675 (b) 3686 (c) 3683 (d) 3678
7. 7.What is the units digit of 3^58? (a) 9 (b) 3 (c) 1 (d) 7
8. 8.What is the units digit of 2^58? (a) 6 (b) 8 (c) 2 (d) 4
9. 9.What is the units digit of 8^12? (a) 4 (b) 8 (c) 2 (d) 6
10. 10.Which of these numbers is divisible by 4? (a) 3419 (b) 3410 (c) 3412 (d) 3414
11. 11.What is the smallest number you must add to 492 to make it divisible by 9? (a) 3 (b) 4 (c) 2 (d) 6
12. 12.What is the units digit of 3^55? (a) 7 (b) 3 (c) 1 (d) 9
13. 13.What is the smallest number you must add to 71 to make it divisible by 4? (a) 3 (b) 1 (c) 2
14. 14.What is the remainder when 712 is divided by 9? (a) 2 (b) 1 (c) 0 (d) 8
15. 15.What is the remainder when 157 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
16. 16.Which of these numbers is divisible by 8? (a) 4135 (b) 4119 (c) 4127 (d) 4120
17. 17.Which of these numbers is divisible by 6? (a) 2875 (b) 2888 (c) 2881 (d) 2880
18. 18.Which of these numbers is divisible by 8? (a) 2203 (b) 2212 (c) 2222 (d) 2208
19. 19.Which of these numbers is divisible by 11? (a) 3515 (b) 3520 (c) 3526 (d) 3533
20. 20.What is the smallest number you must add to 222 to make it divisible by 7? (a) 2 (b) 5 (c) 3 (d) 1
21. 21.Which of these numbers is divisible by 9? (a) 4941 (b) 4955 (c) 4944 (d) 4935
22. 22.What is the remainder when 427 is divided by 9? (a) 4 (b) 3 (c) 5 (d) 6
23. 23.Which of these numbers is divisible by 8? (a) 2004 (b) 2008 (c) 2021 (d) 2009
24. 24.What is the units digit of 3^48? (a) 7 (b) 1 (c) 3 (d) 9
25. 25.What is the smallest number you must add to 637 to make it divisible by 9? (a) 2 (b) 7 (c) 3 (d) 1
26. 26.Which of these numbers is divisible by 4? (a) 3547 (b) 3548 (c) 3549 (d) 3553
27. 27.Which of these numbers is divisible by 11? (a) 1945 (b) 1962 (c) 1951 (d) 1947
28. 28.Which of these numbers is divisible by 9? (a) 641 (b) 631 (c) 656 (d) 639
29. 29.What is the smallest number you must add to 226 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
30. 30.What is the remainder when 301 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
31. 31.What is the remainder when 436 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
32. 32.What is the remainder when 241 is divided by 12? (a) 11 (b) 2 (c) 0 (d) 1
33. 33.What is the units digit of 7^31? (a) 7 (b) 1 (c) 3 (d) 9
34. 34.What is the remainder when 188 is divided by 9? (a) 7 (b) 8 (c) 1 (d) 6
35. 35.What is the smallest number you must add to 102 to make it divisible by 7? (a) 2 (b) 3 (c) 4 (d) 5
36. 36.What is the units digit of 4^18? (a) 7 (b) 6 (c) 4 (d) 5
37. 37.What is the remainder when 162 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
38. 38.What is the units digit of 8^57? (a) 8 (b) 6 (c) 4 (d) 2
39. 39.What is the smallest number you must add to 406 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 4
40. 40.Which of these numbers is divisible by 6? (a) 3515 (b) 3504 (c) 3502 (d) 3506

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