Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 199 · 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.Which of these numbers is divisible by 9? (a) 8085 (b) 8079 (c) 8082 (d) 8097
2. 2.What is the units digit of 9^59? (a) 9 (b) 1 (c) 8 (d) 7
3. 3.What is the units digit of 7^25? (a) 1 (b) 9 (c) 3 (d) 7
4. 4.Which of these numbers is divisible by 3? (a) 2105 (b) 2101 (c) 2108 (d) 2103
5. 5.What is the remainder when 127 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
6. 6.What is the smallest number you must add to 301 to make it divisible by 8? (a) 2 (b) 4 (c) 3 (d) 5
7. 7.Which of these numbers is divisible by 4? (a) 3251 (b) 3247 (c) 3255 (d) 3248
8. 8.Which of these numbers is divisible by 8? (a) 1553 (b) 1571 (c) 1567 (d) 1560
9. 9.What is the units digit of 3^14? (a) 9 (b) 3 (c) 1 (d) 7
10. 10.What is the units digit of 9^34? (a) 0 (b) 1 (c) 2 (d) 9
11. 11.What is the remainder when 439 is divided by 6? (a) 1 (b) 5 (c) 0 (d) 2
12. 12.What is the remainder when 678 is divided by 9? (a) 2 (b) 4 (c) 3 (d) 6
13. 13.What is the smallest number you must add to 208 to make it divisible by 3? (a) 1 (b) 2
14. 14.What is the remainder when 470 is divided by 8? (a) 5 (b) 7 (c) 2 (d) 6
15. 15.Which of these numbers is divisible by 4? (a) 1851 (b) 1847 (c) 1842 (d) 1844
16. 16.What is the units digit of 3^40? (a) 1 (b) 9 (c) 3 (d) 7
17. 17.What is the remainder when 317 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
18. 18.Which of these numbers is divisible by 9? (a) 7402 (b) 7422 (c) 7407 (d) 7410
19. 19.What is the smallest number you must add to 153 to make it divisible by 12? (a) 9 (b) 2 (c) 4 (d) 3
20. 20.What is the units digit of 7^43? (a) 9 (b) 7 (c) 1 (d) 3
21. 21.What is the remainder when 121 is divided by 3? (a) 2 (b) 0 (c) 1
22. 22.What is the units digit of 9^30? (a) 1 (b) 9 (c) 2 (d) 0
23. 23.What is the units digit of 3^19? (a) 9 (b) 3 (c) 7 (d) 1
24. 24.What is the remainder when 153 is divided by 7? (a) 4 (b) 1 (c) 5 (d) 6
25. 25.What is the remainder when 555 is divided by 10? (a) 5 (b) 7 (c) 4 (d) 6
26. 26.What is the smallest number you must add to 95 to make it divisible by 4? (a) 3 (b) 1 (c) 2
27. 27.What is the remainder when 221 is divided by 3? (a) 1 (b) 0 (c) 2
28. 28.What is the smallest number you must add to 277 to make it divisible by 6? (a) 4 (b) 5 (c) 1 (d) 3
29. 29.What is the remainder when 281 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
30. 30.What is the units digit of 7^55? (a) 3 (b) 7 (c) 1 (d) 9
31. 31.What is the remainder when 258 is divided by 10? (a) 8 (b) 2 (c) 9 (d) 7
32. 32.What is the remainder when 390 is divided by 12? (a) 6 (b) 5 (c) 7 (d) 8
33. 33.What is the units digit of 8^58? (a) 8 (b) 6 (c) 2 (d) 4
34. 34.Which of these numbers is divisible by 9? (a) 6696 (b) 6690 (c) 6698 (d) 6706
35. 35.What is the remainder when 283 is divided by 6? (a) 5 (b) 1 (c) 0 (d) 2
36. 36.What is the units digit of 2^53? (a) 6 (b) 4 (c) 8 (d) 2
37. 37.What is the units digit of 9^24? (a) 0 (b) 9 (c) 2 (d) 1
38. 38.What is the units digit of 8^59? (a) 8 (b) 6 (c) 4 (d) 2
39. 39.What is the remainder when 269 is divided by 12? (a) 6 (b) 4 (c) 5 (d) 7
40. 40.What is the smallest number you must add to 262 to make it divisible by 10? (a) 7 (b) 8 (c) 9 (d) 2

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