Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 130 · 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 372 to make it divisible by 7? (a) 1 (b) 6 (c) 4 (d) 5
2. 2.What is the units digit of 7^29? (a) 1 (b) 9 (c) 7 (d) 3
3. 3.What is the units digit of 4^25? (a) 3 (b) 4 (c) 6 (d) 5
4. 4.What is the smallest number you must add to 281 to make it divisible by 9? (a) 7 (b) 8 (c) 2 (d) 6
5. 5.What is the units digit of 3^39? (a) 1 (b) 3 (c) 9 (d) 7
6. 6.What is the smallest number you must add to 173 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 1
7. 7.Which of these numbers is divisible by 6? (a) 1935 (b) 1939 (c) 1930 (d) 1932
8. 8.Which of these numbers is divisible by 8? (a) 4885 (b) 4897 (c) 4890 (d) 4888
9. 9.Which of these numbers is divisible by 8? (a) 6928 (b) 6927 (c) 6931 (d) 6941
10. 10.What is the remainder when 548 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
11. 11.What is the units digit of 9^10? (a) 1 (b) 9 (c) 0 (d) 2
12. 12.What is the remainder when 390 is divided by 11? (a) 4 (b) 7 (c) 6 (d) 5
13. 13.Which of these numbers is divisible by 8? (a) 1170 (b) 1159 (c) 1160 (d) 1167
14. 14.What is the smallest number you must add to 321 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
15. 15.What is the smallest number you must add to 246 to make it divisible by 12? (a) 7 (b) 8 (c) 5 (d) 6
16. 16.What is the units digit of 4^59? (a) 4 (b) 3 (c) 5 (d) 6
17. 17.Which of these numbers is divisible by 4? (a) 751 (b) 748 (c) 745 (d) 753
18. 18.What is the units digit of 9^40? (a) 0 (b) 1 (c) 2 (d) 9
19. 19.Which of these numbers is divisible by 8? (a) 6280 (b) 6283 (c) 6295 (d) 6274
20. 20.What is the units digit of 2^41? (a) 4 (b) 2 (c) 6 (d) 8
21. 21.What is the units digit of 8^51? (a) 6 (b) 4 (c) 8 (d) 2
22. 22.What is the remainder when 187 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
23. 23.What is the remainder when 473 is divided by 12? (a) 6 (b) 4 (c) 5 (d) 7
24. 24.What is the remainder when 88 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
25. 25.What is the units digit of 8^12? (a) 4 (b) 6 (c) 8 (d) 2
26. 26.Which of these numbers is divisible by 3? (a) 1499 (b) 1493 (c) 1494 (d) 1496
27. 27.What is the units digit of 8^30? (a) 8 (b) 6 (c) 2 (d) 4
28. 28.What is the smallest number you must add to 184 to make it divisible by 3? (a) 1 (b) 2
29. 29.Which of these numbers is divisible by 3? (a) 2010 (b) 2009 (c) 2015 (d) 2012
30. 30.What is the remainder when 76 is divided by 5? (a) 2 (b) 1 (c) 0 (d) 4
31. 31.What is the smallest number you must add to 687 to make it divisible by 11? (a) 6 (b) 8 (c) 7 (d) 5
32. 32.What is the units digit of 2^24? (a) 2 (b) 8 (c) 6 (d) 4
33. 33.What is the units digit of 7^15? (a) 7 (b) 1 (c) 9 (d) 3
34. 34.What is the smallest number you must add to 213 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
35. 35.What is the remainder when 149 is divided by 6? (a) 1 (b) 4 (c) 5 (d) 3
36. 36.What is the smallest number you must add to 368 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
37. 37.Which of these numbers is divisible by 9? (a) 4571 (b) 4579 (c) 4584 (d) 4572
38. 38.What is the units digit of 4^36? (a) 6 (b) 5 (c) 7 (d) 4
39. 39.What is the remainder when 516 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 2
40. 40.What is the units digit of 8^40? (a) 8 (b) 4 (c) 6 (d) 2

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