Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 321 · 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 7^10? (a) 3 (b) 7 (c) 1 (d) 9
2. 2.What is the units digit of 2^37? (a) 4 (b) 8 (c) 6 (d) 2
3. 3.What is the remainder when 270 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
4. 4.What is the units digit of 2^13? (a) 4 (b) 8 (c) 2 (d) 6
5. 5.What is the units digit of 4^32? (a) 5 (b) 7 (c) 6 (d) 4
6. 6.Which of these numbers is divisible by 3? (a) 285 (b) 290 (c) 284 (d) 287
7. 7.What is the remainder when 263 is divided by 11? (a) 9 (b) 8 (c) 1 (d) 10
8. 8.What is the smallest number you must add to 155 to make it divisible by 3? (a) 2 (b) 1
9. 9.What is the units digit of 7^26? (a) 3 (b) 9 (c) 1 (d) 7
10. 10.What is the smallest number you must add to 171 to make it divisible by 7? (a) 6 (b) 3 (c) 5 (d) 4
11. 11.What is the units digit of 9^54? (a) 2 (b) 1 (c) 9 (d) 0
12. 12.What is the remainder when 163 is divided by 7? (a) 1 (b) 5 (c) 3 (d) 2
13. 13.What is the remainder when 449 is divided by 8? (a) 1 (b) 2 (c) 0 (d) 7
14. 14.What is the smallest number you must add to 545 to make it divisible by 8? (a) 1 (b) 6 (c) 7 (d) 5
15. 15.Which of these numbers is divisible by 6? (a) 3486 (b) 3481 (c) 3488 (d) 3493
16. 16.What is the smallest number you must add to 445 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 1
17. 17.What is the units digit of 7^56? (a) 7 (b) 1 (c) 3 (d) 9
18. 18.What is the smallest number you must add to 142 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
19. 19.What is the remainder when 222 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
20. 20.What is the units digit of 3^10? (a) 3 (b) 1 (c) 7 (d) 9
21. 21.Which of these numbers is divisible by 8? (a) 6492 (b) 6488 (c) 6485 (d) 6497
22. 22.What is the remainder when 644 is divided by 11? (a) 6 (b) 8 (c) 5 (d) 7
23. 23.Which of these numbers is divisible by 3? (a) 883 (b) 887 (c) 889 (d) 885
24. 24.Which of these numbers is divisible by 6? (a) 3505 (b) 3498 (c) 3501 (d) 3494
25. 25.What is the units digit of 7^20? (a) 1 (b) 9 (c) 3 (d) 7
26. 26.Which of these numbers is divisible by 6? (a) 4958 (b) 4950 (c) 4949 (d) 4954
27. 27.What is the units digit of 4^22? (a) 4 (b) 5 (c) 6 (d) 7
28. 28.Which of these numbers is divisible by 6? (a) 5046 (b) 5054 (c) 5050 (d) 5041
29. 29.What is the smallest number you must add to 214 to make it divisible by 3? (a) 1 (b) 2
30. 30.What is the smallest number you must add to 236 to make it divisible by 3? (a) 2 (b) 1
31. 31.What is the units digit of 2^49? (a) 6 (b) 8 (c) 4 (d) 2
32. 32.Which of these numbers is divisible by 8? (a) 5211 (b) 5206 (c) 5195 (d) 5200
33. 33.What is the smallest number you must add to 431 to make it divisible by 8? (a) 7 (b) 2 (c) 3 (d) 1
34. 34.Which of these numbers is divisible by 4? (a) 835 (b) 828 (c) 825 (d) 831
35. 35.Which of these numbers is divisible by 6? (a) 1732 (b) 1728 (c) 1735 (d) 1727
36. 36.What is the smallest number you must add to 492 to make it divisible by 11? (a) 8 (b) 2 (c) 3 (d) 4
37. 37.Which of these numbers is divisible by 3? (a) 542 (b) 539 (c) 540 (d) 545
38. 38.What is the smallest number you must add to 541 to make it divisible by 11? (a) 8 (b) 10 (c) 9 (d) 2
39. 39.What is the remainder when 342 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
40. 40.What is the remainder when 578 is divided by 10? (a) 2 (b) 7 (c) 9 (d) 8

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