Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 491 · 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 11? (a) 3354 (b) 3333 (c) 3341 (d) 3332
2. 2.What is the remainder when 134 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 2
3. 3.Which of these numbers is divisible by 9? (a) 5981 (b) 5988 (c) 5976 (d) 5974
4. 4.What is the remainder when 803 is divided by 10? (a) 3 (b) 2 (c) 7 (d) 4
5. 5.What is the units digit of 2^14? (a) 4 (b) 2 (c) 8 (d) 6
6. 6.What is the remainder when 213 is divided by 11? (a) 7 (b) 4 (c) 5 (d) 3
7. 7.Which of these numbers is divisible by 8? (a) 1814 (b) 1801 (c) 1808 (d) 1821
8. 8.What is the units digit of 3^23? (a) 3 (b) 7 (c) 9 (d) 1
9. 9.Which of these numbers is divisible by 4? (a) 1068 (b) 1066 (c) 1071 (d) 1075
10. 10.Which of these numbers is divisible by 6? (a) 2463 (b) 2468 (c) 2458 (d) 2460
11. 11.Which of these numbers is divisible by 6? (a) 2808 (b) 2810 (c) 2806 (d) 2818
12. 12.What is the smallest number you must add to 173 to make it divisible by 4? (a) 1 (b) 2 (c) 3
13. 13.Which of these numbers is divisible by 6? (a) 5212 (b) 5202 (c) 5197 (d) 5203
14. 14.What is the smallest number you must add to 733 to make it divisible by 10? (a) 6 (b) 7 (c) 8 (d) 3
15. 15.What is the smallest number you must add to 41 to make it divisible by 4? (a) 2 (b) 1 (c) 3
16. 16.What is the smallest number you must add to 617 to make it divisible by 12? (a) 5 (b) 8 (c) 6 (d) 7
17. 17.What is the units digit of 8^31? (a) 2 (b) 4 (c) 8 (d) 6
18. 18.Which of these numbers is divisible by 4? (a) 1522 (b) 1524 (c) 1529 (d) 1527
19. 19.What is the units digit of 7^15? (a) 3 (b) 1 (c) 7 (d) 9
20. 20.What is the units digit of 9^28? (a) 2 (b) 9 (c) 1 (d) 0
21. 21.What is the units digit of 4^19? (a) 5 (b) 3 (c) 4 (d) 6
22. 22.What is the remainder when 110 is divided by 9? (a) 3 (b) 2 (c) 7 (d) 1
23. 23.What is the remainder when 423 is divided by 11? (a) 4 (b) 6 (c) 7 (d) 5
24. 24.What is the remainder when 202 is divided by 3? (a) 2 (b) 0 (c) 1
25. 25.What is the units digit of 9^46? (a) 9 (b) 2 (c) 0 (d) 1
26. 26.What is the remainder when 806 is divided by 11? (a) 3 (b) 8 (c) 2 (d) 4
27. 27.What is the units digit of 8^51? (a) 8 (b) 4 (c) 6 (d) 2
28. 28.Which of these numbers is divisible by 3? (a) 2308 (b) 2306 (c) 2312 (d) 2307
29. 29.What is the units digit of 3^48? (a) 9 (b) 3 (c) 1 (d) 7
30. 30.What is the remainder when 299 is divided by 12? (a) 11 (b) 9 (c) 1 (d) 10
31. 31.What is the remainder when 93 is divided by 7? (a) 3 (b) 1 (c) 5 (d) 2
32. 32.What is the smallest number you must add to 475 to make it divisible by 7? (a) 2 (b) 1 (c) 3 (d) 6
33. 33.What is the smallest number you must add to 247 to make it divisible by 7? (a) 2 (b) 4 (c) 6 (d) 5
34. 34.What is the remainder when 97 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
35. 35.What is the remainder when 108 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
36. 36.What is the units digit of 7^48? (a) 9 (b) 3 (c) 1 (d) 7
37. 37.Which of these numbers is divisible by 8? (a) 6234 (b) 6247 (c) 6229 (d) 6232
38. 38.What is the units digit of 9^55? (a) 7 (b) 1 (c) 9 (d) 8
39. 39.What is the smallest number you must add to 529 to make it divisible by 12? (a) 11 (b) 1 (c) 9 (d) 10
40. 40.What is the smallest number you must add to 188 to make it divisible by 11? (a) 9 (b) 1 (c) 10 (d) 8

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