Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 590 · 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 smallest number you must add to 182 to make it divisible by 8? (a) 3 (b) 1 (c) 2 (d) 6
2. 2.Which of these numbers is divisible by 11? (a) 4265 (b) 4257 (c) 4272 (d) 4251
3. 3.Which of these numbers is divisible by 9? (a) 3969 (b) 3979 (c) 3972 (d) 3967
4. 4.What is the units digit of 9^21? (a) 8 (b) 9 (c) 7 (d) 1
5. 5.What is the units digit of 7^10? (a) 9 (b) 7 (c) 3 (d) 1
6. 6.Which of these numbers is divisible by 11? (a) 3706 (b) 3719 (c) 3707 (d) 3711
7. 7.What is the units digit of 8^49? (a) 8 (b) 2 (c) 6 (d) 4
8. 8.What is the units digit of 9^24? (a) 2 (b) 0 (c) 1 (d) 9
9. 9.Which of these numbers is divisible by 6? (a) 3899 (b) 3887 (c) 3888 (d) 3890
10. 10.What is the units digit of 4^41? (a) 6 (b) 5 (c) 3 (d) 4
11. 11.What is the smallest number you must add to 401 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 1
12. 12.What is the remainder when 46 is divided by 3? (a) 1 (b) 2 (c) 0
13. 13.What is the units digit of 7^19? (a) 7 (b) 1 (c) 3 (d) 9
14. 14.Which of these numbers is divisible by 8? (a) 1522 (b) 1518 (c) 1512 (d) 1506
15. 15.Which of these numbers is divisible by 3? (a) 434 (b) 431 (c) 429 (d) 428
16. 16.Which of these numbers is divisible by 4? (a) 503 (b) 500 (c) 497 (d) 505
17. 17.Which of these numbers is divisible by 4? (a) 1887 (b) 1880 (c) 1882 (d) 1878
18. 18.What is the units digit of 4^31? (a) 5 (b) 6 (c) 4 (d) 3
19. 19.What is the smallest number you must add to 230 to make it divisible by 4? (a) 3 (b) 2 (c) 1
20. 20.What is the units digit of 4^39? (a) 3 (b) 5 (c) 4 (d) 6
21. 21.What is the smallest number you must add to 578 to make it divisible by 10? (a) 1 (b) 8 (c) 2 (d) 3
22. 22.What is the remainder when 510 is divided by 9? (a) 5 (b) 7 (c) 6 (d) 3
23. 23.What is the remainder when 170 is divided by 8? (a) 2 (b) 1 (c) 3 (d) 6
24. 24.What is the remainder when 62 is divided by 3? (a) 1 (b) 2 (c) 0
25. 25.What is the smallest number you must add to 333 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
26. 26.Which of these numbers is divisible by 6? (a) 2444 (b) 2441 (c) 2453 (d) 2442
27. 27.What is the smallest number you must add to 566 to make it divisible by 9? (a) 3 (b) 1 (c) 2 (d) 8
28. 28.What is the units digit of 9^44? (a) 9 (b) 2 (c) 0 (d) 1
29. 29.What is the smallest number you must add to 181 to make it divisible by 3? (a) 2 (b) 1
30. 30.What is the units digit of 9^22? (a) 2 (b) 9 (c) 1 (d) 0
31. 31.What is the units digit of 3^14? (a) 3 (b) 1 (c) 9 (d) 7
32. 32.Which of these numbers is divisible by 11? (a) 2742 (b) 2750 (c) 2757 (d) 2768
33. 33.What is the remainder when 453 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
34. 34.What is the remainder when 123 is divided by 9? (a) 7 (b) 5 (c) 3 (d) 6
35. 35.What is the units digit of 8^32? (a) 8 (b) 4 (c) 6 (d) 2
36. 36.What is the remainder when 253 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
37. 37.What is the smallest number you must add to 609 to make it divisible by 10? (a) 3 (b) 2 (c) 1 (d) 9
38. 38.What is the units digit of 9^18? (a) 2 (b) 0 (c) 1 (d) 9
39. 39.What is the remainder when 361 is divided by 6? (a) 2 (b) 0 (c) 5 (d) 1
40. 40.What is the units digit of 3^15? (a) 1 (b) 9 (c) 3 (d) 7

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