Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 223 · 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) 4058 (b) 4066 (c) 4056 (d) 4051
2. 2.What is the units digit of 4^48? (a) 4 (b) 5 (c) 7 (d) 6
3. 3.What is the smallest number you must add to 255 to make it divisible by 10? (a) 7 (b) 6 (c) 4 (d) 5
4. 4.What is the remainder when 381 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 0
5. 5.What is the units digit of 3^16? (a) 9 (b) 7 (c) 3 (d) 1
6. 6.Which of these numbers is divisible by 8? (a) 5490 (b) 5497 (c) 5488 (d) 5487
7. 7.What is the smallest number you must add to 456 to make it divisible by 11? (a) 7 (b) 8 (c) 5 (d) 6
8. 8.What is the units digit of 9^22? (a) 0 (b) 9 (c) 1 (d) 2
9. 9.Which of these numbers is divisible by 11? (a) 894 (b) 902 (c) 920 (d) 907
10. 10.What is the units digit of 3^59? (a) 3 (b) 7 (c) 9 (d) 1
11. 11.What is the units digit of 7^43? (a) 7 (b) 9 (c) 1 (d) 3
12. 12.What is the smallest number you must add to 84 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
13. 13.What is the units digit of 7^41? (a) 9 (b) 3 (c) 1 (d) 7
14. 14.What is the remainder when 351 is divided by 7? (a) 0 (b) 2 (c) 1 (d) 6
15. 15.What is the smallest number you must add to 93 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
16. 16.Which of these numbers is divisible by 8? (a) 2717 (b) 2710 (c) 2721 (d) 2712
17. 17.Which of these numbers is divisible by 8? (a) 5014 (b) 5002 (c) 5017 (d) 5008
18. 18.Which of these numbers is divisible by 6? (a) 4713 (b) 4716 (c) 4725 (d) 4717
19. 19.What is the units digit of 8^32? (a) 2 (b) 6 (c) 8 (d) 4
20. 20.What is the smallest number you must add to 177 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
21. 21.Which of these numbers is divisible by 4? (a) 732 (b) 739 (c) 734 (d) 729
22. 22.Which of these numbers is divisible by 3? (a) 1825 (b) 1831 (c) 1827 (d) 1828
23. 23.What is the smallest number you must add to 257 to make it divisible by 10? (a) 4 (b) 7 (c) 2 (d) 3
24. 24.Which of these numbers is divisible by 4? (a) 768 (b) 769 (c) 767 (d) 773
25. 25.What is the units digit of 4^22? (a) 5 (b) 7 (c) 6 (d) 4
26. 26.Which of these numbers is divisible by 9? (a) 5886 (b) 5890 (c) 5900 (d) 5884
27. 27.What is the units digit of 3^41? (a) 1 (b) 9 (c) 7 (d) 3
28. 28.What is the units digit of 3^29? (a) 3 (b) 9 (c) 1 (d) 7
29. 29.Which of these numbers is divisible by 6? (a) 2404 (b) 2396 (c) 2394 (d) 2391
30. 30.What is the remainder when 719 is divided by 9? (a) 1 (b) 6 (c) 8 (d) 7
31. 31.What is the smallest number you must add to 287 to make it divisible by 12? (a) 3 (b) 11 (c) 1 (d) 2
32. 32.What is the units digit of 7^15? (a) 1 (b) 9 (c) 7 (d) 3
33. 33.Which of these numbers is divisible by 11? (a) 4334 (b) 4337 (c) 4328 (d) 4350
34. 34.Which of these numbers is divisible by 4? (a) 2768 (b) 2771 (c) 2775 (d) 2767
35. 35.Which of these numbers is divisible by 4? (a) 2109 (b) 2107 (c) 2104 (d) 2103
36. 36.What is the units digit of 4^37? (a) 5 (b) 4 (c) 6 (d) 3
37. 37.What is the smallest number you must add to 483 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 2
38. 38.Which of these numbers is divisible by 8? (a) 7067 (b) 7052 (c) 7056 (d) 7063
39. 39.What is the units digit of 2^52? (a) 2 (b) 6 (c) 8 (d) 4
40. 40.What is the units digit of 3^22? (a) 9 (b) 3 (c) 7 (d) 1

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