Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 846 · 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 2^23? (a) 2 (b) 8 (c) 4 (d) 6
2. 2.What is the remainder when 768 is divided by 10? (a) 2 (b) 9 (c) 8 (d) 7
3. 3.What is the smallest number you must add to 344 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
4. 4.What is the units digit of 4^14? (a) 5 (b) 6 (c) 4 (d) 7
5. 5.Which of these numbers is divisible by 4? (a) 3332 (b) 3331 (c) 3334 (d) 3339
6. 6.What is the units digit of 4^42? (a) 4 (b) 7 (c) 6 (d) 5
7. 7.Which of these numbers is divisible by 4? (a) 1751 (b) 1752 (c) 1753 (d) 1759
8. 8.What is the remainder when 184 is divided by 3? (a) 2 (b) 0 (c) 1
9. 9.What is the smallest number you must add to 134 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
10. 10.What is the smallest number you must add to 578 to make it divisible by 10? (a) 1 (b) 8 (c) 2 (d) 3
11. 11.What is the remainder when 134 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
12. 12.What is the remainder when 238 is divided by 8? (a) 5 (b) 7 (c) 6 (d) 2
13. 13.What is the remainder when 224 is divided by 6? (a) 4 (b) 1 (c) 2 (d) 3
14. 14.Which of these numbers is divisible by 6? (a) 3372 (b) 3369 (c) 3383 (d) 3377
15. 15.What is the smallest number you must add to 782 to make it divisible by 11? (a) 8 (b) 1 (c) 10 (d) 9
16. 16.What is the remainder when 224 is divided by 3? (a) 1 (b) 0 (c) 2
17. 17.Which of these numbers is divisible by 4? (a) 525 (b) 524 (c) 523 (d) 530
18. 18.What is the units digit of 2^50? (a) 2 (b) 8 (c) 4 (d) 6
19. 19.What is the smallest number you must add to 640 to make it divisible by 12? (a) 7 (b) 8 (c) 9 (d) 4
20. 20.What is the remainder when 99 is divided by 7? (a) 2 (b) 0 (c) 1 (d) 6
21. 21.What is the remainder when 113 is divided by 7? (a) 1 (b) 6 (c) 0 (d) 2
22. 22.What is the units digit of 8^30? (a) 8 (b) 2 (c) 4 (d) 6
23. 23.What is the smallest number you must add to 213 to make it divisible by 4? (a) 2 (b) 3 (c) 1
24. 24.What is the smallest number you must add to 65 to make it divisible by 3? (a) 2 (b) 1
25. 25.What is the remainder when 53 is divided by 3? (a) 1 (b) 0 (c) 2
26. 26.What is the units digit of 7^12? (a) 3 (b) 1 (c) 9 (d) 7
27. 27.What is the units digit of 4^52? (a) 4 (b) 6 (c) 7 (d) 5
28. 28.What is the smallest number you must add to 521 to make it divisible by 10? (a) 9 (b) 1 (c) 8 (d) 7
29. 29.What is the smallest number you must add to 334 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
30. 30.Which of these numbers is divisible by 11? (a) 9384 (b) 9380 (c) 9369 (d) 9372
31. 31.What is the remainder when 374 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
32. 32.What is the units digit of 8^12? (a) 8 (b) 6 (c) 4 (d) 2
33. 33.What is the remainder when 129 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
34. 34.What is the units digit of 9^40? (a) 1 (b) 2 (c) 0 (d) 9
35. 35.What is the smallest number you must add to 687 to make it divisible by 12? (a) 10 (b) 3 (c) 8 (d) 9
36. 36.Which of these numbers is divisible by 6? (a) 1581 (b) 1574 (c) 1588 (d) 1578
37. 37.What is the smallest number you must add to 115 to make it divisible by 4? (a) 2 (b) 3 (c) 1
38. 38.What is the units digit of 7^58? (a) 3 (b) 7 (c) 9 (d) 1
39. 39.What is the smallest number you must add to 257 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
40. 40.What is the smallest number you must add to 116 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4

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