Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 537 · 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 8^18? (a) 8 (b) 4 (c) 2 (d) 6
2. 2.Which of these numbers is divisible by 6? (a) 3934 (b) 3927 (c) 3920 (d) 3924
3. 3.What is the smallest number you must add to 368 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
4. 4.Which of these numbers is divisible by 11? (a) 462 (b) 476 (c) 457 (d) 465
5. 5.What is the smallest number you must add to 209 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 5
6. 6.What is the units digit of 7^42? (a) 9 (b) 7 (c) 1 (d) 3
7. 7.Which of these numbers is divisible by 8? (a) 7090 (b) 7111 (c) 7102 (d) 7096
8. 8.Which of these numbers is divisible by 3? (a) 823 (b) 819 (c) 821 (d) 818
9. 9.Which of these numbers is divisible by 8? (a) 5381 (b) 5386 (c) 5398 (d) 5384
10. 10.What is the units digit of 7^24? (a) 1 (b) 7 (c) 3 (d) 9
11. 11.What is the remainder when 438 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 3
12. 12.What is the smallest number you must add to 401 to make it divisible by 7? (a) 6 (b) 2 (c) 5 (d) 4
13. 13.Which of these numbers is divisible by 8? (a) 5964 (b) 5968 (c) 5977 (d) 5972
14. 14.What is the units digit of 2^34? (a) 2 (b) 6 (c) 8 (d) 4
15. 15.What is the units digit of 3^17? (a) 9 (b) 1 (c) 7 (d) 3
16. 16.Which of these numbers is divisible by 4? (a) 1363 (b) 1356 (c) 1359 (d) 1353
17. 17.What is the units digit of 2^31? (a) 4 (b) 8 (c) 6 (d) 2
18. 18.Which of these numbers is divisible by 6? (a) 4356 (b) 4358 (c) 4351 (d) 4365
19. 19.What is the remainder when 694 is divided by 10? (a) 5 (b) 3 (c) 4 (d) 6
20. 20.What is the units digit of 7^25? (a) 1 (b) 3 (c) 9 (d) 7
21. 21.What is the remainder when 667 is divided by 12? (a) 5 (b) 8 (c) 6 (d) 7
22. 22.Which of these numbers is divisible by 11? (a) 3682 (b) 3663 (c) 3659 (d) 3666
23. 23.What is the remainder when 235 is divided by 7? (a) 6 (b) 4 (c) 3 (d) 5
24. 24.Which of these numbers is divisible by 6? (a) 382 (b) 378 (c) 374 (d) 385
25. 25.What is the units digit of 9^10? (a) 0 (b) 2 (c) 9 (d) 1
26. 26.What is the units digit of 2^15? (a) 8 (b) 2 (c) 6 (d) 4
27. 27.Which of these numbers is divisible by 9? (a) 5057 (b) 5065 (c) 5049 (d) 5044
28. 28.Which of these numbers is divisible by 3? (a) 944 (b) 945 (c) 947 (d) 949
29. 29.What is the remainder when 225 is divided by 10? (a) 7 (b) 6 (c) 5 (d) 4
30. 30.Which of these numbers is divisible by 11? (a) 3388 (b) 3384 (c) 3408 (d) 3397
31. 31.Which of these numbers is divisible by 11? (a) 6007 (b) 5998 (c) 6006 (d) 6025
32. 32.What is the smallest number you must add to 356 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
33. 33.What is the remainder when 254 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
34. 34.What is the remainder when 194 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
35. 35.What is the units digit of 7^17? (a) 1 (b) 9 (c) 3 (d) 7
36. 36.What is the units digit of 9^30? (a) 9 (b) 1 (c) 2 (d) 0
37. 37.What is the units digit of 4^29? (a) 6 (b) 5 (c) 4 (d) 3
38. 38.What is the smallest number you must add to 458 to make it divisible by 7? (a) 3 (b) 6 (c) 5 (d) 4
39. 39.Which of these numbers is divisible by 11? (a) 2675 (b) 2671 (c) 2694 (d) 2673
40. 40.Which of these numbers is divisible by 3? (a) 768 (b) 770 (c) 773 (d) 766

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