Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 725 · 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 4? (a) 3104 (b) 3103 (c) 3107 (d) 3111
2. 2.What is the smallest number you must add to 499 to make it divisible by 7? (a) 2 (b) 6 (c) 5 (d) 4
3. 3.What is the remainder when 86 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
4. 4.Which of these numbers is divisible by 4? (a) 258 (b) 256 (c) 263 (d) 253
5. 5.What is the remainder when 861 is divided by 12? (a) 9 (b) 10 (c) 8 (d) 3
6. 6.What is the smallest number you must add to 301 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
7. 7.What is the smallest number you must add to 165 to make it divisible by 8? (a) 3 (b) 2 (c) 5 (d) 4
8. 8.What is the remainder when 953 is divided by 12? (a) 5 (b) 4 (c) 6 (d) 7
9. 9.What is the units digit of 4^43? (a) 3 (b) 4 (c) 6 (d) 5
10. 10.Which of these numbers is divisible by 9? (a) 7013 (b) 7007 (c) 6995 (d) 7002
11. 11.What is the remainder when 62 is divided by 3? (a) 0 (b) 1 (c) 2
12. 12.What is the units digit of 7^26? (a) 3 (b) 1 (c) 9 (d) 7
13. 13.Which of these numbers is divisible by 9? (a) 7222 (b) 7209 (c) 7210 (d) 7203
14. 14.What is the smallest number you must add to 82 to make it divisible by 3? (a) 1 (b) 2
15. 15.Which of these numbers is divisible by 11? (a) 5310 (b) 5316 (c) 5325 (d) 5313
16. 16.What is the units digit of 9^39? (a) 9 (b) 7 (c) 1 (d) 8
17. 17.Which of these numbers is divisible by 3? (a) 2232 (b) 2236 (c) 2233 (d) 2230
18. 18.What is the remainder when 147 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
19. 19.What is the units digit of 7^18? (a) 3 (b) 7 (c) 9 (d) 1
20. 20.What is the smallest number you must add to 164 to make it divisible by 8? (a) 5 (b) 6 (c) 3 (d) 4
21. 21.What is the units digit of 4^19? (a) 5 (b) 4 (c) 3 (d) 6
22. 22.What is the remainder when 53 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
23. 23.What is the units digit of 8^15? (a) 8 (b) 6 (c) 4 (d) 2
24. 24.What is the smallest number you must add to 118 to make it divisible by 4? (a) 1 (b) 3 (c) 2
25. 25.What is the units digit of 2^16? (a) 4 (b) 8 (c) 6 (d) 2
26. 26.What is the remainder when 221 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
27. 27.What is the smallest number you must add to 394 to make it divisible by 9? (a) 2 (b) 1 (c) 7 (d) 3
28. 28.What is the smallest number you must add to 296 to make it divisible by 7? (a) 2 (b) 6 (c) 4 (d) 5
29. 29.Which of these numbers is divisible by 9? (a) 3436 (b) 3417 (c) 3426 (d) 3420
30. 30.What is the smallest number you must add to 187 to make it divisible by 9? (a) 3 (b) 2 (c) 7 (d) 1
31. 31.Which of these numbers is divisible by 4? (a) 2473 (b) 2481 (c) 2479 (d) 2476
32. 32.Which of these numbers is divisible by 3? (a) 2099 (b) 2097 (c) 2095 (d) 2101
33. 33.What is the remainder when 434 is divided by 12? (a) 1 (b) 10 (c) 3 (d) 2
34. 34.Which of these numbers is divisible by 4? (a) 1838 (b) 1842 (c) 1847 (d) 1840
35. 35.What is the units digit of 7^51? (a) 1 (b) 3 (c) 9 (d) 7
36. 36.What is the remainder when 373 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
37. 37.What is the smallest number you must add to 559 to make it divisible by 9? (a) 1 (b) 8 (c) 7 (d) 6
38. 38.Which of these numbers is divisible by 6? (a) 428 (b) 424 (c) 437 (d) 426
39. 39.What is the remainder when 531 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 2
40. 40.What is the smallest number you must add to 610 to make it divisible by 11? (a) 8 (b) 5 (c) 7 (d) 6

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