Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 259 · 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 253 to make it divisible by 4? (a) 1 (b) 2 (c) 3
2. 2.What is the units digit of 8^39? (a) 2 (b) 6 (c) 8 (d) 4
3. 3.Which of these numbers is divisible by 9? (a) 4157 (b) 4148 (c) 4162 (d) 4149
4. 4.What is the units digit of 3^24? (a) 9 (b) 1 (c) 7 (d) 3
5. 5.Which of these numbers is divisible by 3? (a) 1070 (b) 1063 (c) 1065 (d) 1067
6. 6.Which of these numbers is divisible by 8? (a) 4838 (b) 4826 (c) 4822 (d) 4824
7. 7.Which of these numbers is divisible by 6? (a) 2757 (b) 2745 (c) 2751 (d) 2748
8. 8.What is the smallest number you must add to 409 to make it divisible by 9? (a) 7 (b) 5 (c) 4 (d) 6
9. 9.What is the units digit of 7^57? (a) 7 (b) 1 (c) 9 (d) 3
10. 10.What is the units digit of 3^27? (a) 9 (b) 7 (c) 3 (d) 1
11. 11.What is the remainder when 61 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
12. 12.Which of these numbers is divisible by 6? (a) 3504 (b) 3503 (c) 3505 (d) 3514
13. 13.What is the smallest number you must add to 267 to make it divisible by 4? (a) 1 (b) 3 (c) 2
14. 14.Which of these numbers is divisible by 3? (a) 1537 (b) 1531 (c) 1533 (d) 1534
15. 15.Which of these numbers is divisible by 6? (a) 1277 (b) 1272 (c) 1283 (d) 1271
16. 16.Which of these numbers is divisible by 6? (a) 1226 (b) 1221 (c) 1224 (d) 1231
17. 17.What is the remainder when 95 is divided by 8? (a) 1 (b) 5 (c) 6 (d) 7
18. 18.What is the smallest number you must add to 146 to make it divisible by 3? (a) 1 (b) 2
19. 19.Which of these numbers is divisible by 8? (a) 4206 (b) 4214 (c) 4200 (d) 4194
20. 20.What is the remainder when 557 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 3
21. 21.What is the units digit of 9^26? (a) 2 (b) 0 (c) 1 (d) 9
22. 22.What is the smallest number you must add to 227 to make it divisible by 9? (a) 2 (b) 7 (c) 6 (d) 8
23. 23.Which of these numbers is divisible by 6? (a) 4250 (b) 4237 (c) 4242 (d) 4247
24. 24.What is the units digit of 7^47? (a) 1 (b) 9 (c) 3 (d) 7
25. 25.What is the units digit of 3^53? (a) 1 (b) 7 (c) 9 (d) 3
26. 26.What is the smallest number you must add to 237 to make it divisible by 4? (a) 2 (b) 1 (c) 3
27. 27.What is the units digit of 2^54? (a) 4 (b) 8 (c) 6 (d) 2
28. 28.What is the remainder when 178 is divided by 10? (a) 8 (b) 9 (c) 7 (d) 2
29. 29.Which of these numbers is divisible by 8? (a) 3192 (b) 3194 (c) 3201 (d) 3186
30. 30.What is the remainder when 146 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
31. 31.Which of these numbers is divisible by 8? (a) 5008 (b) 5011 (c) 5023 (d) 5001
32. 32.What is the smallest number you must add to 230 to make it divisible by 8? (a) 6 (b) 3 (c) 1 (d) 2
33. 33.What is the units digit of 9^12? (a) 9 (b) 2 (c) 0 (d) 1
34. 34.What is the units digit of 3^40? (a) 9 (b) 7 (c) 3 (d) 1
35. 35.What is the smallest number you must add to 581 to make it divisible by 12? (a) 5 (b) 6 (c) 7 (d) 8
36. 36.What is the units digit of 2^27? (a) 4 (b) 2 (c) 6 (d) 8
37. 37.Which of these numbers is divisible by 3? (a) 1067 (b) 1062 (c) 1064 (d) 1060
38. 38.What is the remainder when 104 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2
39. 39.What is the units digit of 9^11? (a) 9 (b) 8 (c) 7 (d) 1
40. 40.What is the remainder when 181 is divided by 10? (a) 1 (b) 0 (c) 9 (d) 2

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