Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 354 · 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 8? (a) 5304 (b) 5301 (c) 5305 (d) 5316
2. 2.What is the smallest number you must add to 540 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
3. 3.What is the units digit of 4^17? (a) 4 (b) 3 (c) 5 (d) 6
4. 4.What is the remainder when 178 is divided by 7? (a) 2 (b) 5 (c) 3 (d) 4
5. 5.What is the smallest number you must add to 179 to make it divisible by 12? (a) 2 (b) 1 (c) 3 (d) 11
6. 6.Which of these numbers is divisible by 4? (a) 203 (b) 205 (c) 200 (d) 199
7. 7.Which of these numbers is divisible by 3? (a) 1198 (b) 1201 (c) 1197 (d) 1196
8. 8.What is the units digit of 8^44? (a) 4 (b) 6 (c) 2 (d) 8
9. 9.What is the units digit of 7^15? (a) 7 (b) 1 (c) 9 (d) 3
10. 10.Which of these numbers is divisible by 11? (a) 1000 (b) 979 (c) 973 (d) 985
11. 11.What is the units digit of 3^54? (a) 9 (b) 1 (c) 7 (d) 3
12. 12.What is the units digit of 2^46? (a) 8 (b) 6 (c) 4 (d) 2
13. 13.Which of these numbers is divisible by 3? (a) 583 (b) 589 (c) 585 (d) 587
14. 14.Which of these numbers is divisible by 3? (a) 308 (b) 311 (c) 314 (d) 309
15. 15.What is the units digit of 2^50? (a) 8 (b) 6 (c) 2 (d) 4
16. 16.What is the remainder when 524 is divided by 12? (a) 9 (b) 8 (c) 7 (d) 4
17. 17.What is the remainder when 313 is divided by 11? (a) 4 (b) 7 (c) 5 (d) 6
18. 18.What is the smallest number you must add to 122 to make it divisible by 3? (a) 1 (b) 2
19. 19.What is the remainder when 169 is divided by 3? (a) 0 (b) 1 (c) 2
20. 20.Which of these numbers is divisible by 4? (a) 3407 (b) 3410 (c) 3408 (d) 3413
21. 21.What is the units digit of 8^37? (a) 2 (b) 6 (c) 8 (d) 4
22. 22.What is the units digit of 3^48? (a) 3 (b) 7 (c) 9 (d) 1
23. 23.What is the units digit of 2^32? (a) 8 (b) 6 (c) 2 (d) 4
24. 24.What is the smallest number you must add to 75 to make it divisible by 4? (a) 1 (b) 3 (c) 2
25. 25.What is the units digit of 4^43? (a) 5 (b) 4 (c) 6 (d) 3
26. 26.What is the units digit of 2^13? (a) 8 (b) 2 (c) 4 (d) 6
27. 27.What is the units digit of 2^57? (a) 2 (b) 8 (c) 4 (d) 6
28. 28.Which of these numbers is divisible by 3? (a) 873 (b) 874 (c) 878 (d) 871
29. 29.What is the smallest number you must add to 397 to make it divisible by 9? (a) 7 (b) 8 (c) 6 (d) 1
30. 30.What is the remainder when 106 is divided by 9? (a) 2 (b) 6 (c) 8 (d) 7
31. 31.What is the remainder when 458 is divided by 8? (a) 3 (b) 2 (c) 1 (d) 6
32. 32.What is the remainder when 476 is divided by 6? (a) 2 (b) 1 (c) 3 (d) 4
33. 33.What is the smallest number you must add to 137 to make it divisible by 10? (a) 2 (b) 4 (c) 3 (d) 7
34. 34.Which of these numbers is divisible by 9? (a) 2069 (b) 2081 (c) 2075 (d) 2070
35. 35.What is the remainder when 191 is divided by 3? (a) 0 (b) 1 (c) 2
36. 36.What is the remainder when 219 is divided by 11? (a) 1 (b) 9 (c) 8 (d) 10
37. 37.Which of these numbers is divisible by 6? (a) 1841 (b) 1844 (c) 1849 (d) 1842
38. 38.Which of these numbers is divisible by 4? (a) 1162 (b) 1160 (c) 1165 (d) 1157
39. 39.What is the remainder when 420 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
40. 40.What is the smallest number you must add to 727 to make it divisible by 12? (a) 6 (b) 7 (c) 5 (d) 4

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