Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 689 · 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 2^57? (a) 4 (b) 8 (c) 2 (d) 6
2. 2.What is the smallest number you must add to 463 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 1
3. 3.What is the smallest number you must add to 141 to make it divisible by 11? (a) 2 (b) 9 (c) 3 (d) 1
4. 4.What is the smallest number you must add to 625 to make it divisible by 9? (a) 4 (b) 7 (c) 6 (d) 5
5. 5.Which of these numbers is divisible by 4? (a) 967 (b) 957 (c) 961 (d) 960
6. 6.What is the smallest number you must add to 447 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3
7. 7.What is the remainder when 575 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
8. 8.What is the remainder when 214 is divided by 3? (a) 0 (b) 1 (c) 2
9. 9.Which of these numbers is divisible by 3? (a) 1909 (b) 1911 (c) 1916 (d) 1912
10. 10.What is the remainder when 317 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
11. 11.Which of these numbers is divisible by 9? (a) 8016 (b) 7995 (c) 8001 (d) 8009
12. 12.What is the smallest number you must add to 134 to make it divisible by 3? (a) 2 (b) 1
13. 13.What is the units digit of 8^23? (a) 4 (b) 2 (c) 8 (d) 6
14. 14.What is the units digit of 3^43? (a) 9 (b) 7 (c) 3 (d) 1
15. 15.What is the units digit of 2^40? (a) 6 (b) 4 (c) 8 (d) 2
16. 16.What is the units digit of 7^33? (a) 3 (b) 9 (c) 7 (d) 1
17. 17.What is the units digit of 4^41? (a) 5 (b) 6 (c) 3 (d) 4
18. 18.What is the smallest number you must add to 604 to make it divisible by 9? (a) 7 (b) 6 (c) 1 (d) 8
19. 19.What is the units digit of 2^33? (a) 6 (b) 2 (c) 4 (d) 8
20. 20.What is the units digit of 7^46? (a) 7 (b) 1 (c) 9 (d) 3
21. 21.What is the smallest number you must add to 376 to make it divisible by 9? (a) 1 (b) 3 (c) 2 (d) 7
22. 22.What is the smallest number you must add to 222 to make it divisible by 4? (a) 2 (b) 1 (c) 3
23. 23.Which of these numbers is divisible by 6? (a) 5226 (b) 5227 (c) 5224 (d) 5234
24. 24.What is the units digit of 3^24? (a) 3 (b) 1 (c) 9 (d) 7
25. 25.Which of these numbers is divisible by 11? (a) 4062 (b) 4054 (c) 4038 (d) 4048
26. 26.What is the units digit of 4^54? (a) 4 (b) 6 (c) 5 (d) 7
27. 27.Which of these numbers is divisible by 6? (a) 4249 (b) 4265 (c) 4254 (d) 4255
28. 28.Which of these numbers is divisible by 6? (a) 2419 (b) 2425 (c) 2424 (d) 2431
29. 29.What is the units digit of 2^42? (a) 4 (b) 2 (c) 8 (d) 6
30. 30.What is the units digit of 7^27? (a) 7 (b) 9 (c) 1 (d) 3
31. 31.What is the remainder when 130 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
32. 32.What is the smallest number you must add to 315 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
33. 33.What is the units digit of 3^40? (a) 7 (b) 9 (c) 1 (d) 3
34. 34.What is the remainder when 209 is divided by 3? (a) 0 (b) 1 (c) 2
35. 35.What is the units digit of 9^16? (a) 1 (b) 2 (c) 9 (d) 0
36. 36.What is the remainder when 103 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
37. 37.What is the remainder when 253 is divided by 7? (a) 1 (b) 0 (c) 6 (d) 2
38. 38.Which of these numbers is divisible by 3? (a) 593 (b) 591 (c) 596 (d) 590
39. 39.What is the units digit of 3^58? (a) 1 (b) 9 (c) 7 (d) 3
40. 40.What is the remainder when 157 is divided by 12? (a) 2 (b) 0 (c) 11 (d) 1

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