Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 412 · 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 remainder when 440 is divided by 6? (a) 2 (b) 4 (c) 1 (d) 3
2. 2.What is the units digit of 2^11? (a) 4 (b) 8 (c) 6 (d) 2
3. 3.What is the remainder when 297 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
4. 4.What is the units digit of 7^32? (a) 3 (b) 9 (c) 7 (d) 1
5. 5.What is the smallest number you must add to 348 to make it divisible by 11? (a) 4 (b) 3 (c) 5 (d) 7
6. 6.What is the remainder when 256 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 0
7. 7.Which of these numbers is divisible by 4? (a) 2919 (b) 2920 (c) 2926 (d) 2922
8. 8.What is the remainder when 594 is divided by 8? (a) 6 (b) 2 (c) 1 (d) 3
9. 9.Which of these numbers is divisible by 3? (a) 2039 (b) 2036 (c) 2032 (d) 2034
10. 10.What is the smallest number you must add to 297 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
11. 11.What is the smallest number you must add to 214 to make it divisible by 3? (a) 2 (b) 1
12. 12.Which of these numbers is divisible by 4? (a) 1607 (b) 1600 (c) 1598 (d) 1603
13. 13.What is the smallest number you must add to 556 to make it divisible by 7? (a) 3 (b) 6 (c) 5 (d) 4
14. 14.What is the remainder when 356 is divided by 12? (a) 9 (b) 7 (c) 4 (d) 8
15. 15.What is the smallest number you must add to 379 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 1
16. 16.What is the units digit of 3^12? (a) 3 (b) 1 (c) 7 (d) 9
17. 17.What is the units digit of 8^56? (a) 2 (b) 6 (c) 4 (d) 8
18. 18.What is the units digit of 3^16? (a) 9 (b) 7 (c) 1 (d) 3
19. 19.What is the smallest number you must add to 93 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 2
20. 20.What is the remainder when 626 is divided by 10? (a) 7 (b) 4 (c) 5 (d) 6
21. 21.What is the smallest number you must add to 510 to make it divisible by 9? (a) 6 (b) 2 (c) 3 (d) 4
22. 22.What is the remainder when 83 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
23. 23.What is the units digit of 2^51? (a) 8 (b) 2 (c) 6 (d) 4
24. 24.Which of these numbers is divisible by 4? (a) 1059 (b) 1051 (c) 1052 (d) 1053
25. 25.What is the units digit of 4^18? (a) 6 (b) 4 (c) 5 (d) 7
26. 26.Which of these numbers is divisible by 3? (a) 2629 (b) 2631 (c) 2635 (d) 2633
27. 27.What is the remainder when 490 is divided by 11? (a) 5 (b) 6 (c) 8 (d) 7
28. 28.What is the units digit of 2^56? (a) 2 (b) 8 (c) 6 (d) 4
29. 29.Which of these numbers is divisible by 6? (a) 2856 (b) 2861 (c) 2865 (d) 2851
30. 30.What is the units digit of 7^12? (a) 3 (b) 7 (c) 9 (d) 1
31. 31.What is the smallest number you must add to 253 to make it divisible by 7? (a) 4 (b) 5 (c) 1 (d) 6
32. 32.What is the remainder when 259 is divided by 8? (a) 3 (b) 5 (c) 2 (d) 4
33. 33.Which of these numbers is divisible by 6? (a) 668 (b) 666 (c) 677 (d) 665
34. 34.What is the remainder when 116 is divided by 3? (a) 0 (b) 1 (c) 2
35. 35.What is the units digit of 2^44? (a) 4 (b) 6 (c) 2 (d) 8
36. 36.What is the remainder when 511 is divided by 8? (a) 1 (b) 5 (c) 6 (d) 7
37. 37.Which of these numbers is divisible by 8? (a) 1679 (b) 1664 (c) 1659 (d) 1666
38. 38.Which of these numbers is divisible by 3? (a) 206 (b) 202 (c) 208 (d) 204
39. 39.What is the units digit of 9^59? (a) 7 (b) 8 (c) 1 (d) 9
40. 40.Which of these numbers is divisible by 3? (a) 1568 (b) 1566 (c) 1571 (d) 1565

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