Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 738 · 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 85 to make it divisible by 3? (a) 2 (b) 1
2. 2.What is the remainder when 38 is divided by 3? (a) 1 (b) 2 (c) 0
3. 3.What is the units digit of 4^43? (a) 6 (b) 3 (c) 5 (d) 4
4. 4.What is the units digit of 3^20? (a) 7 (b) 9 (c) 3 (d) 1
5. 5.Which of these numbers is divisible by 3? (a) 1131 (b) 1130 (c) 1132 (d) 1136
6. 6.Which of these numbers is divisible by 6? (a) 4202 (b) 4208 (c) 4200 (d) 4196
7. 7.What is the remainder when 191 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
8. 8.What is the smallest number you must add to 491 to make it divisible by 10? (a) 7 (b) 9 (c) 1 (d) 8
9. 9.What is the units digit of 2^20? (a) 8 (b) 6 (c) 4 (d) 2
10. 10.What is the units digit of 2^12? (a) 8 (b) 2 (c) 6 (d) 4
11. 11.What is the remainder when 235 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
12. 12.What is the remainder when 224 is divided by 3? (a) 1 (b) 0 (c) 2
13. 13.What is the units digit of 4^49? (a) 6 (b) 3 (c) 4 (d) 5
14. 14.Which of these numbers is divisible by 11? (a) 3451 (b) 3429 (c) 3432 (d) 3442
15. 15.What is the remainder when 115 is divided by 3? (a) 2 (b) 1 (c) 0
16. 16.Which of these numbers is divisible by 9? (a) 3486 (b) 3474 (c) 3477 (d) 3469
17. 17.What is the remainder when 194 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 2
18. 18.What is the remainder when 291 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
19. 19.What is the remainder when 53 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
20. 20.What is the remainder when 838 is divided by 11? (a) 2 (b) 9 (c) 1 (d) 3
21. 21.Which of these numbers is divisible by 9? (a) 1585 (b) 1584 (c) 1598 (d) 1579
22. 22.What is the units digit of 9^14? (a) 9 (b) 0 (c) 2 (d) 1
23. 23.Which of these numbers is divisible by 4? (a) 1916 (b) 1922 (c) 1914 (d) 1917
24. 24.Which of these numbers is divisible by 4? (a) 1004 (b) 1005 (c) 1001 (d) 1010
25. 25.What is the units digit of 7^20? (a) 1 (b) 7 (c) 3 (d) 9
26. 26.Which of these numbers is divisible by 3? (a) 209 (b) 204 (c) 206 (d) 203
27. 27.What is the remainder when 287 is divided by 8? (a) 5 (b) 6 (c) 1 (d) 7
28. 28.Which of these numbers is divisible by 11? (a) 8770 (b) 8792 (c) 8780 (d) 8778
29. 29.What is the smallest number you must add to 412 to make it divisible by 8? (a) 6 (b) 5 (c) 4 (d) 3
30. 30.What is the units digit of 2^28? (a) 8 (b) 6 (c) 2 (d) 4
31. 31.Which of these numbers is divisible by 8? (a) 2674 (b) 2667 (c) 2685 (d) 2672
32. 32.What is the units digit of 3^15? (a) 9 (b) 7 (c) 1 (d) 3
33. 33.Which of these numbers is divisible by 9? (a) 4058 (b) 4041 (c) 4035 (d) 4042
34. 34.What is the remainder when 580 is divided by 12? (a) 4 (b) 5 (c) 8 (d) 3
35. 35.What is the remainder when 245 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
36. 36.What is the units digit of 2^17? (a) 6 (b) 2 (c) 8 (d) 4
37. 37.Which of these numbers is divisible by 4? (a) 3099 (b) 3093 (c) 3092 (d) 3089
38. 38.What is the remainder when 277 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
39. 39.What is the units digit of 3^50? (a) 7 (b) 3 (c) 1 (d) 9
40. 40.What is the smallest number you must add to 598 to make it divisible by 12? (a) 10 (b) 1 (c) 3 (d) 2

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