Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 878 · 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 177 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
2. 2.What is the units digit of 3^40? (a) 3 (b) 9 (c) 1 (d) 7
3. 3.What is the remainder when 150 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
4. 4.What is the smallest number you must add to 284 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
5. 5.Which of these numbers is divisible by 11? (a) 8371 (b) 8392 (c) 8379 (d) 8365
6. 6.What is the smallest number you must add to 439 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
7. 7.What is the units digit of 2^17? (a) 8 (b) 6 (c) 2 (d) 4
8. 8.What is the remainder when 241 is divided by 3? (a) 0 (b) 1 (c) 2
9. 9.What is the smallest number you must add to 52 to make it divisible by 3? (a) 1 (b) 2
10. 10.Which of these numbers is divisible by 9? (a) 5750 (b) 5751 (c) 5765 (d) 5756
11. 11.What is the remainder when 651 is divided by 11? (a) 3 (b) 9 (c) 2 (d) 1
12. 12.Which of these numbers is divisible by 3? (a) 2477 (b) 2480 (c) 2478 (d) 2482
13. 13.What is the smallest number you must add to 130 to make it divisible by 3? (a) 1 (b) 2
14. 14.Which of these numbers is divisible by 6? (a) 733 (b) 732 (c) 727 (d) 743
15. 15.What is the remainder when 241 is divided by 5? (a) 2 (b) 1 (c) 0 (d) 4
16. 16.What is the remainder when 94 is divided by 6? (a) 3 (b) 5 (c) 2 (d) 4
17. 17.What is the remainder when 326 is divided by 12? (a) 3 (b) 10 (c) 1 (d) 2
18. 18.What is the units digit of 9^54? (a) 0 (b) 2 (c) 1 (d) 9
19. 19.Which of these numbers is divisible by 8? (a) 3863 (b) 3851 (c) 3848 (d) 3843
20. 20.Which of these numbers is divisible by 3? (a) 2113 (b) 2119 (c) 2115 (d) 2116
21. 21.Which of these numbers is divisible by 9? (a) 6310 (b) 6297 (c) 6304 (d) 6300
22. 22.Which of these numbers is divisible by 11? (a) 8536 (b) 8544 (c) 8531 (d) 8557
23. 23.What is the units digit of 7^22? (a) 1 (b) 9 (c) 3 (d) 7
24. 24.What is the remainder when 79 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
25. 25.What is the remainder when 439 is divided by 7? (a) 6 (b) 4 (c) 2 (d) 5
26. 26.What is the remainder when 457 is divided by 9? (a) 8 (b) 6 (c) 2 (d) 7
27. 27.Which of these numbers is divisible by 9? (a) 6281 (b) 6259 (c) 6264 (d) 6267
28. 28.What is the remainder when 57 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
29. 29.What is the smallest number you must add to 94 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
30. 30.What is the remainder when 69 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
31. 31.What is the smallest number you must add to 115 to make it divisible by 3? (a) 1 (b) 2
32. 32.What is the units digit of 8^40? (a) 4 (b) 6 (c) 2 (d) 8
33. 33.What is the smallest number you must add to 226 to make it divisible by 3? (a) 2 (b) 1
34. 34.Which of these numbers is divisible by 3? (a) 1333 (b) 1330 (c) 1327 (d) 1329
35. 35.What is the units digit of 7^29? (a) 1 (b) 9 (c) 3 (d) 7
36. 36.Which of these numbers is divisible by 8? (a) 4914 (b) 4906 (c) 4912 (d) 4923
37. 37.What is the units digit of 4^44? (a) 5 (b) 4 (c) 6 (d) 7
38. 38.What is the units digit of 3^23? (a) 3 (b) 1 (c) 7 (d) 9
39. 39.What is the smallest number you must add to 109 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
40. 40.Which of these numbers is divisible by 11? (a) 3762 (b) 3752 (c) 3766 (d) 3774

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