Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 545 · 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 3^31? (a) 3 (b) 7 (c) 9 (d) 1
2. 2.What is the smallest number you must add to 586 to make it divisible by 9? (a) 8 (b) 7 (c) 6 (d) 1
3. 3.Which of these numbers is divisible by 8? (a) 6530 (b) 6514 (c) 6521 (d) 6520
4. 4.What is the units digit of 4^24? (a) 4 (b) 6 (c) 7 (d) 5
5. 5.What is the smallest number you must add to 67 to make it divisible by 3? (a) 1 (b) 2
6. 6.Which of these numbers is divisible by 3? (a) 421 (b) 428 (c) 423 (d) 424
7. 7.What is the units digit of 8^50? (a) 6 (b) 4 (c) 2 (d) 8
8. 8.What is the remainder when 178 is divided by 9? (a) 6 (b) 2 (c) 7 (d) 8
9. 9.Which of these numbers is divisible by 4? (a) 3051 (b) 3048 (c) 3046 (d) 3054
10. 10.Which of these numbers is divisible by 11? (a) 6831 (b) 6837 (c) 6844 (d) 6821
11. 11.What is the units digit of 3^32? (a) 3 (b) 1 (c) 7 (d) 9
12. 12.What is the smallest number you must add to 318 to make it divisible by 9? (a) 7 (b) 5 (c) 3 (d) 6
13. 13.Which of these numbers is divisible by 8? (a) 6888 (b) 6889 (c) 6903 (d) 6887
14. 14.What is the units digit of 9^24? (a) 9 (b) 2 (c) 1 (d) 0
15. 15.Which of these numbers is divisible by 6? (a) 3095 (b) 3101 (c) 3096 (d) 3103
16. 16.What is the smallest number you must add to 281 to make it divisible by 9? (a) 6 (b) 2 (c) 7 (d) 8
17. 17.What is the remainder when 305 is divided by 11? (a) 7 (b) 9 (c) 3 (d) 8
18. 18.What is the smallest number you must add to 661 to make it divisible by 12? (a) 9 (b) 1 (c) 11 (d) 10
19. 19.What is the remainder when 323 is divided by 8? (a) 5 (b) 2 (c) 3 (d) 4
20. 20.What is the smallest number you must add to 756 to make it divisible by 11? (a) 4 (b) 2 (c) 8 (d) 3
21. 21.What is the smallest number you must add to 91 to make it divisible by 3? (a) 2 (b) 1
22. 22.What is the remainder when 300 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
23. 23.Which of these numbers is divisible by 4? (a) 1387 (b) 1391 (c) 1383 (d) 1384
24. 24.What is the smallest number you must add to 236 to make it divisible by 3? (a) 2 (b) 1
25. 25.What is the remainder when 247 is divided by 6? (a) 2 (b) 1 (c) 0 (d) 5
26. 26.What is the remainder when 97 is divided by 8? (a) 0 (b) 1 (c) 2 (d) 7
27. 27.Which of these numbers is divisible by 11? (a) 9689 (b) 9696 (c) 9703 (d) 9691
28. 28.What is the smallest number you must add to 428 to make it divisible by 9? (a) 5 (b) 3 (c) 6 (d) 4
29. 29.What is the smallest number you must add to 293 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 6
30. 30.What is the smallest number you must add to 147 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
31. 31.What is the smallest number you must add to 328 to make it divisible by 9? (a) 6 (b) 4 (c) 5 (d) 7
32. 32.What is the smallest number you must add to 264 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
33. 33.What is the remainder when 282 is divided by 10? (a) 2 (b) 8 (c) 1 (d) 3
34. 34.What is the units digit of 2^37? (a) 2 (b) 4 (c) 6 (d) 8
35. 35.What is the units digit of 7^12? (a) 3 (b) 1 (c) 7 (d) 9
36. 36.What is the units digit of 7^11? (a) 7 (b) 3 (c) 9 (d) 1
37. 37.Which of these numbers is divisible by 11? (a) 3146 (b) 3147 (c) 3166 (d) 3143
38. 38.What is the units digit of 9^52? (a) 1 (b) 0 (c) 2 (d) 9
39. 39.What is the units digit of 3^10? (a) 7 (b) 9 (c) 3 (d) 1
40. 40.Which of these numbers is divisible by 9? (a) 5029 (b) 5031 (c) 5039 (d) 5044

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