Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 345 · 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 4^11? (a) 5 (b) 6 (c) 3 (d) 4
2. 2.What is the units digit of 7^57? (a) 3 (b) 9 (c) 1 (d) 7
3. 3.What is the smallest number you must add to 79 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 1
4. 4.Which of these numbers is divisible by 3? (a) 2074 (b) 2081 (c) 2076 (d) 2078
5. 5.What is the units digit of 2^25? (a) 4 (b) 6 (c) 8 (d) 2
6. 6.What is the units digit of 9^52? (a) 2 (b) 9 (c) 1 (d) 0
7. 7.What is the units digit of 9^14? (a) 2 (b) 0 (c) 1 (d) 9
8. 8.What is the remainder when 393 is divided by 12? (a) 8 (b) 3 (c) 10 (d) 9
9. 9.What is the remainder when 97 is divided by 6? (a) 5 (b) 2 (c) 1 (d) 0
10. 10.Which of these numbers is divisible by 3? (a) 2098 (b) 2104 (c) 2100 (d) 2101
11. 11.Which of these numbers is divisible by 6? (a) 1163 (b) 1168 (c) 1164 (d) 1173
12. 12.What is the smallest number you must add to 493 to make it divisible by 12? (a) 10 (b) 11 (c) 1 (d) 9
13. 13.What is the remainder when 454 is divided by 11? (a) 4 (b) 8 (c) 2 (d) 3
14. 14.Which of these numbers is divisible by 9? (a) 4036 (b) 4032 (c) 4027 (d) 4043
15. 15.What is the smallest number you must add to 688 to make it divisible by 11? (a) 4 (b) 5 (c) 7 (d) 6
16. 16.What is the units digit of 9^12? (a) 0 (b) 9 (c) 1 (d) 2
17. 17.What is the remainder when 208 is divided by 7? (a) 2 (b) 5 (c) 6 (d) 4
18. 18.What is the units digit of 7^16? (a) 7 (b) 1 (c) 3 (d) 9
19. 19.What is the remainder when 173 is divided by 8? (a) 5 (b) 6 (c) 3 (d) 4
20. 20.What is the remainder when 314 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
21. 21.Which of these numbers is divisible by 4? (a) 923 (b) 924 (c) 931 (d) 927
22. 22.What is the remainder when 923 is divided by 12? (a) 10 (b) 9 (c) 11 (d) 1
23. 23.What is the units digit of 4^41? (a) 4 (b) 6 (c) 3 (d) 5
24. 24.What is the units digit of 8^16? (a) 4 (b) 6 (c) 8 (d) 2
25. 25.Which of these numbers is divisible by 8? (a) 4239 (b) 4232 (c) 4244 (d) 4226
26. 26.What is the remainder when 317 is divided by 9? (a) 3 (b) 1 (c) 2 (d) 7
27. 27.What is the units digit of 4^59? (a) 3 (b) 5 (c) 6 (d) 4
28. 28.Which of these numbers is divisible by 9? (a) 5462 (b) 5464 (c) 5454 (d) 5450
29. 29.Which of these numbers is divisible by 9? (a) 7656 (b) 7669 (c) 7659 (d) 7663
30. 30.What is the smallest number you must add to 665 to make it divisible by 12? (a) 8 (b) 7 (c) 6 (d) 5
31. 31.What is the units digit of 8^13? (a) 2 (b) 8 (c) 4 (d) 6
32. 32.Which of these numbers is divisible by 9? (a) 1016 (b) 1021 (c) 1001 (d) 1008
33. 33.What is the smallest number you must add to 439 to make it divisible by 11? (a) 1 (b) 10 (c) 3 (d) 2
34. 34.Which of these numbers is divisible by 8? (a) 3897 (b) 3888 (c) 3883 (d) 3890
35. 35.What is the smallest number you must add to 735 to make it divisible by 12? (a) 10 (b) 3 (c) 9 (d) 8
36. 36.What is the smallest number you must add to 798 to make it divisible by 10? (a) 3 (b) 1 (c) 2 (d) 8
37. 37.What is the smallest number you must add to 199 to make it divisible by 10? (a) 3 (b) 9 (c) 2 (d) 1
38. 38.What is the units digit of 8^39? (a) 4 (b) 6 (c) 2 (d) 8
39. 39.What is the smallest number you must add to 121 to make it divisible by 6? (a) 5 (b) 4 (c) 1 (d) 3
40. 40.What is the units digit of 8^59? (a) 4 (b) 8 (c) 6 (d) 2

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