Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 179 · 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 3^47? (a) 9 (b) 3 (c) 1 (d) 7
2. 2.What is the remainder when 346 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 3
3. 3.What is the remainder when 112 is divided by 3? (a) 0 (b) 2 (c) 1
4. 4.What is the smallest number you must add to 401 to make it divisible by 12? (a) 6 (b) 5 (c) 7 (d) 8
5. 5.What is the remainder when 582 is divided by 11? (a) 9 (b) 1 (c) 8 (d) 10
6. 6.What is the smallest number you must add to 438 to make it divisible by 9? (a) 3 (b) 6 (c) 2 (d) 4
7. 7.Which of these numbers is divisible by 6? (a) 1241 (b) 1245 (c) 1252 (d) 1242
8. 8.What is the smallest number you must add to 651 to make it divisible by 12? (a) 10 (b) 8 (c) 3 (d) 9
9. 9.Which of these numbers is divisible by 8? (a) 3367 (b) 3354 (c) 3360 (d) 3370
10. 10.Which of these numbers is divisible by 6? (a) 2034 (b) 2044 (c) 2029 (d) 2036
11. 11.What is the remainder when 362 is divided by 7? (a) 6 (b) 2 (c) 5 (d) 4
12. 12.Which of these numbers is divisible by 4? (a) 861 (b) 860 (c) 865 (d) 858
13. 13.Which of these numbers is divisible by 9? (a) 2026 (b) 2041 (c) 2025 (d) 2022
14. 14.Which of these numbers is divisible by 8? (a) 520 (b) 522 (c) 533 (d) 517
15. 15.What is the remainder when 345 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
16. 16.What is the remainder when 155 is divided by 6? (a) 5 (b) 4 (c) 1 (d) 3
17. 17.What is the units digit of 9^55? (a) 1 (b) 7 (c) 9 (d) 8
18. 18.What is the units digit of 4^23? (a) 5 (b) 6 (c) 3 (d) 4
19. 19.What is the units digit of 8^50? (a) 8 (b) 6 (c) 2 (d) 4
20. 20.Which of these numbers is divisible by 6? (a) 3812 (b) 3823 (c) 3821 (d) 3816
21. 21.Which of these numbers is divisible by 11? (a) 2574 (b) 2586 (c) 2564 (d) 2579
22. 22.Which of these numbers is divisible by 6? (a) 940 (b) 930 (c) 925 (d) 935
23. 23.What is the smallest number you must add to 193 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
24. 24.What is the remainder when 281 is divided by 8? (a) 0 (b) 2 (c) 1 (d) 7
25. 25.What is the remainder when 332 is divided by 10? (a) 1 (b) 8 (c) 2 (d) 3
26. 26.What is the units digit of 7^17? (a) 9 (b) 7 (c) 3 (d) 1
27. 27.What is the units digit of 9^40? (a) 1 (b) 2 (c) 9 (d) 0
28. 28.Which of these numbers is divisible by 8? (a) 5113 (b) 5098 (c) 5104 (d) 5108
29. 29.Which of these numbers is divisible by 9? (a) 520 (b) 504 (c) 497 (d) 512
30. 30.What is the remainder when 217 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
31. 31.What is the remainder when 335 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 1
32. 32.Which of these numbers is divisible by 9? (a) 941 (b) 955 (c) 950 (d) 945
33. 33.Which of these numbers is divisible by 6? (a) 2652 (b) 2659 (c) 2655 (d) 2651
34. 34.Which of these numbers is divisible by 6? (a) 2049 (b) 2040 (c) 2036 (d) 2041
35. 35.What is the remainder when 179 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
36. 36.What is the smallest number you must add to 868 to make it divisible by 11? (a) 10 (b) 3 (c) 2 (d) 1
37. 37.What is the units digit of 7^55? (a) 7 (b) 1 (c) 9 (d) 3
38. 38.Which of these numbers is divisible by 11? (a) 4113 (b) 4115 (c) 4114 (d) 4129
39. 39.What is the units digit of 7^10? (a) 9 (b) 1 (c) 7 (d) 3
40. 40.What is the smallest number you must add to 183 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 5

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