Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 667 · 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 309 to make it divisible by 8? (a) 5 (b) 2 (c) 4 (d) 3
2. 2.What is the smallest number you must add to 42 to make it divisible by 4? (a) 1 (b) 2 (c) 3
3. 3.What is the remainder when 484 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 5
4. 4.Which of these numbers is divisible by 9? (a) 3195 (b) 3196 (c) 3205 (d) 3190
5. 5.Which of these numbers is divisible by 4? (a) 1913 (b) 1917 (c) 1922 (d) 1916
6. 6.What is the smallest number you must add to 149 to make it divisible by 3? (a) 1 (b) 2
7. 7.What is the units digit of 2^58? (a) 2 (b) 8 (c) 6 (d) 4
8. 8.What is the smallest number you must add to 510 to make it divisible by 9? (a) 6 (b) 3 (c) 4 (d) 2
9. 9.What is the smallest number you must add to 294 to make it divisible by 4? (a) 1 (b) 2 (c) 3
10. 10.Which of these numbers is divisible by 9? (a) 5340 (b) 5337 (c) 5349 (d) 5331
11. 11.What is the smallest number you must add to 394 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 1
12. 12.Which of these numbers is divisible by 6? (a) 4926 (b) 4923 (c) 4937 (d) 4928
13. 13.What is the remainder when 332 is divided by 6? (a) 4 (b) 1 (c) 2 (d) 3
14. 14.What is the remainder when 90 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
15. 15.Which of these numbers is divisible by 9? (a) 743 (b) 726 (c) 729 (d) 732
16. 16.Which of these numbers is divisible by 6? (a) 4612 (b) 4608 (c) 4616 (d) 4604
17. 17.What is the remainder when 573 is divided by 11? (a) 2 (b) 0 (c) 1 (d) 10
18. 18.What is the remainder when 247 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
19. 19.Which of these numbers is divisible by 3? (a) 2449 (b) 2453 (c) 2451 (d) 2456
20. 20.Which of these numbers is divisible by 4? (a) 3077 (b) 3082 (c) 3080 (d) 3087
21. 21.Which of these numbers is divisible by 6? (a) 5310 (b) 5308 (c) 5312 (d) 5319
22. 22.What is the remainder when 880 is divided by 12? (a) 3 (b) 4 (c) 8 (d) 5
23. 23.What is the remainder when 167 is divided by 10? (a) 8 (b) 3 (c) 7 (d) 6
24. 24.Which of these numbers is divisible by 6? (a) 445 (b) 435 (c) 438 (d) 440
25. 25.What is the units digit of 3^12? (a) 7 (b) 9 (c) 3 (d) 1
26. 26.What is the remainder when 224 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 3
27. 27.Which of these numbers is divisible by 11? (a) 6378 (b) 6399 (c) 6381 (d) 6380
28. 28.What is the remainder when 954 is divided by 12? (a) 7 (b) 5 (c) 6 (d) 8
29. 29.What is the smallest number you must add to 157 to make it divisible by 4? (a) 3 (b) 1 (c) 2
30. 30.Which of these numbers is divisible by 4? (a) 1726 (b) 1721 (c) 1720 (d) 1717
31. 31.What is the remainder when 87 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
32. 32.Which of these numbers is divisible by 3? (a) 2044 (b) 2038 (c) 2040 (d) 2041
33. 33.What is the smallest number you must add to 274 to make it divisible by 4? (a) 3 (b) 2 (c) 1
34. 34.What is the units digit of 8^40? (a) 6 (b) 2 (c) 8 (d) 4
35. 35.What is the units digit of 9^13? (a) 1 (b) 7 (c) 9 (d) 8
36. 36.What is the remainder when 211 is divided by 8? (a) 5 (b) 3 (c) 4 (d) 2
37. 37.Which of these numbers is divisible by 11? (a) 6874 (b) 6853 (c) 6845 (d) 6863
38. 38.What is the remainder when 233 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
39. 39.What is the remainder when 535 is divided by 8? (a) 1 (b) 5 (c) 6 (d) 7
40. 40.What is the units digit of 7^23? (a) 9 (b) 1 (c) 7 (d) 3

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