Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 13 · 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 47 to make it divisible by 4? (a) 3 (b) 2 (c) 1
2. 2.What is the units digit of 8^57? (a) 2 (b) 6 (c) 8 (d) 4
3. 3.What is the units digit of 9^50? (a) 9 (b) 0 (c) 1 (d) 2
4. 4.What is the units digit of 7^18? (a) 3 (b) 1 (c) 7 (d) 9
5. 5.Which of these numbers is divisible by 9? (a) 1896 (b) 1881 (c) 1878 (d) 1882
6. 6.What is the units digit of 4^45? (a) 6 (b) 5 (c) 3 (d) 4
7. 7.What is the units digit of 2^56? (a) 8 (b) 4 (c) 2 (d) 6
8. 8.Which of these numbers is divisible by 6? (a) 2906 (b) 2898 (c) 2897 (d) 2899
9. 9.What is the units digit of 2^54? (a) 2 (b) 6 (c) 8 (d) 4
10. 10.What is the remainder when 290 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
11. 11.Which of these numbers is divisible by 4? (a) 3505 (b) 3504 (c) 3503 (d) 3511
12. 12.What is the remainder when 365 is divided by 11? (a) 1 (b) 9 (c) 2 (d) 3
13. 13.What is the units digit of 3^27? (a) 7 (b) 1 (c) 9 (d) 3
14. 14.What is the smallest number you must add to 118 to make it divisible by 8? (a) 6 (b) 2 (c) 3 (d) 1
15. 15.What is the smallest number you must add to 164 to make it divisible by 3? (a) 2 (b) 1
16. 16.What is the remainder when 277 is divided by 11? (a) 3 (b) 9 (c) 1 (d) 2
17. 17.Which of these numbers is divisible by 3? (a) 1447 (b) 1444 (c) 1443 (d) 1441
18. 18.What is the remainder when 121 is divided by 3? (a) 0 (b) 2 (c) 1
19. 19.What is the remainder when 425 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 1
20. 20.What is the remainder when 776 is divided by 11? (a) 7 (b) 8 (c) 6 (d) 5
21. 21.What is the units digit of 2^12? (a) 4 (b) 6 (c) 8 (d) 2
22. 22.What is the units digit of 7^11? (a) 3 (b) 7 (c) 1 (d) 9
23. 23.What is the units digit of 2^36? (a) 2 (b) 8 (c) 6 (d) 4
24. 24.What is the units digit of 2^57? (a) 4 (b) 2 (c) 8 (d) 6
25. 25.What is the smallest number you must add to 132 to make it divisible by 9? (a) 4 (b) 3 (c) 6 (d) 2
26. 26.What is the remainder when 372 is divided by 10? (a) 2 (b) 8 (c) 3 (d) 1
27. 27.What is the smallest number you must add to 485 to make it divisible by 10? (a) 7 (b) 4 (c) 5 (d) 6
28. 28.Which of these numbers is divisible by 6? (a) 1687 (b) 1683 (c) 1677 (d) 1680
29. 29.What is the remainder when 598 is divided by 12? (a) 9 (b) 2 (c) 10 (d) 11
30. 30.Which of these numbers is divisible by 8? (a) 6226 (b) 6234 (c) 6224 (d) 6218
31. 31.What is the units digit of 2^33? (a) 6 (b) 4 (c) 8 (d) 2
32. 32.What is the units digit of 3^13? (a) 3 (b) 9 (c) 7 (d) 1
33. 33.What is the remainder when 628 is divided by 9? (a) 8 (b) 6 (c) 2 (d) 7
34. 34.What is the remainder when 189 is divided by 8? (a) 5 (b) 3 (c) 6 (d) 4
35. 35.Which of these numbers is divisible by 6? (a) 3175 (b) 3171 (c) 3165 (d) 3168
36. 36.What is the smallest number you must add to 262 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
37. 37.What is the units digit of 4^36? (a) 6 (b) 5 (c) 4 (d) 7
38. 38.Which of these numbers is divisible by 6? (a) 611 (b) 615 (c) 612 (d) 621
39. 39.What is the smallest number you must add to 514 to make it divisible by 8? (a) 2 (b) 5 (c) 7 (d) 6
40. 40.What is the remainder when 558 is divided by 12? (a) 6 (b) 5 (c) 8 (d) 7

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