Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 185 · 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 4^42? (a) 5 (b) 6 (c) 4 (d) 7
2. 2.What is the units digit of 2^11? (a) 4 (b) 8 (c) 6 (d) 2
3. 3.Which of these numbers is divisible by 3? (a) 1598 (b) 1593 (c) 1595 (d) 1592
4. 4.What is the remainder when 211 is divided by 11? (a) 9 (b) 2 (c) 1 (d) 3
5. 5.What is the units digit of 2^29? (a) 6 (b) 8 (c) 4 (d) 2
6. 6.What is the remainder when 227 is divided by 6? (a) 3 (b) 1 (c) 4 (d) 5
7. 7.What is the remainder when 176 is divided by 3? (a) 2 (b) 1 (c) 0
8. 8.What is the remainder when 733 is divided by 12? (a) 2 (b) 0 (c) 11 (d) 1
9. 9.What is the smallest number you must add to 379 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 4
10. 10.Which of these numbers is divisible by 6? (a) 2993 (b) 2983 (c) 2982 (d) 2980
11. 11.Which of these numbers is divisible by 4? (a) 491 (b) 493 (c) 488 (d) 485
12. 12.What is the remainder when 281 is divided by 8? (a) 2 (b) 7 (c) 1 (d) 0
13. 13.What is the smallest number you must add to 433 to make it divisible by 12? (a) 11 (b) 10 (c) 9 (d) 1
14. 14.Which of these numbers is divisible by 4? (a) 3367 (b) 3371 (c) 3373 (d) 3368
15. 15.What is the smallest number you must add to 232 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
16. 16.What is the remainder when 376 is divided by 10? (a) 5 (b) 6 (c) 7 (d) 4
17. 17.What is the units digit of 7^46? (a) 3 (b) 9 (c) 1 (d) 7
18. 18.What is the smallest number you must add to 399 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
19. 19.Which of these numbers is divisible by 4? (a) 589 (b) 595 (c) 592 (d) 599
20. 20.What is the smallest number you must add to 190 to make it divisible by 4? (a) 2 (b) 3 (c) 1
21. 21.Which of these numbers is divisible by 6? (a) 4064 (b) 4079 (c) 4073 (d) 4068
22. 22.What is the remainder when 115 is divided by 3? (a) 2 (b) 1 (c) 0
23. 23.What is the smallest number you must add to 700 to make it divisible by 12? (a) 4 (b) 8 (c) 7 (d) 9
24. 24.What is the smallest number you must add to 150 to make it divisible by 8? (a) 2 (b) 6 (c) 3 (d) 1
25. 25.Which of these numbers is divisible by 4? (a) 651 (b) 655 (c) 652 (d) 657
26. 26.What is the units digit of 4^49? (a) 4 (b) 6 (c) 5 (d) 3
27. 27.What is the smallest number you must add to 289 to make it divisible by 4? (a) 2 (b) 1 (c) 3
28. 28.Which of these numbers is divisible by 6? (a) 1409 (b) 1414 (c) 1402 (d) 1404
29. 29.What is the remainder when 272 is divided by 7? (a) 4 (b) 5 (c) 1 (d) 6
30. 30.Which of these numbers is divisible by 3? (a) 1138 (b) 1140 (c) 1144 (d) 1141
31. 31.What is the smallest number you must add to 603 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
32. 32.What is the smallest number you must add to 484 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 1
33. 33.Which of these numbers is divisible by 9? (a) 6392 (b) 6399 (c) 6410 (d) 6400
34. 34.What is the units digit of 9^14? (a) 2 (b) 1 (c) 9 (d) 0
35. 35.What is the remainder when 405 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 1
36. 36.What is the units digit of 2^19? (a) 2 (b) 8 (c) 6 (d) 4
37. 37.What is the remainder when 106 is divided by 8? (a) 1 (b) 6 (c) 3 (d) 2
38. 38.What is the remainder when 548 is divided by 9? (a) 6 (b) 1 (c) 8 (d) 7
39. 39.Which of these numbers is divisible by 6? (a) 1809 (b) 1814 (c) 1821 (d) 1812
40. 40.What is the units digit of 3^19? (a) 1 (b) 3 (c) 7 (d) 9

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