Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 79 · 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 remainder when 269 is divided by 7? (a) 3 (b) 4 (c) 2 (d) 5
2. 2.What is the remainder when 113 is divided by 10? (a) 4 (b) 2 (c) 3 (d) 7
3. 3.What is the remainder when 94 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
4. 4.Which of these numbers is divisible by 6? (a) 1908 (b) 1919 (c) 1911 (d) 1905
5. 5.What is the units digit of 7^55? (a) 7 (b) 1 (c) 9 (d) 3
6. 6.Which of these numbers is divisible by 6? (a) 4537 (b) 4536 (c) 4546 (d) 4533
7. 7.What is the smallest number you must add to 121 to make it divisible by 7? (a) 5 (b) 2 (c) 4 (d) 6
8. 8.What is the smallest number you must add to 818 to make it divisible by 11? (a) 7 (b) 6 (c) 8 (d) 4
9. 9.What is the units digit of 4^21? (a) 6 (b) 4 (c) 3 (d) 5
10. 10.Which of these numbers is divisible by 3? (a) 2631 (b) 2633 (c) 2635 (d) 2630
11. 11.What is the remainder when 212 is divided by 3? (a) 0 (b) 2 (c) 1
12. 12.Which of these numbers is divisible by 4? (a) 2591 (b) 2594 (c) 2588 (d) 2585
13. 13.What is the units digit of 3^27? (a) 1 (b) 9 (c) 3 (d) 7
14. 14.Which of these numbers is divisible by 4? (a) 403 (b) 405 (c) 399 (d) 400
15. 15.Which of these numbers is divisible by 4? (a) 2182 (b) 2175 (c) 2176 (d) 2179
16. 16.Which of these numbers is divisible by 3? (a) 1481 (b) 1477 (c) 1476 (d) 1475
17. 17.What is the remainder when 208 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
18. 18.What is the remainder when 190 is divided by 3? (a) 0 (b) 2 (c) 1
19. 19.Which of these numbers is divisible by 3? (a) 1341 (b) 1340 (c) 1345 (d) 1342
20. 20.Which of these numbers is divisible by 6? (a) 455 (b) 458 (c) 450 (d) 446
21. 21.What is the remainder when 347 is divided by 12? (a) 1 (b) 10 (c) 9 (d) 11
22. 22.What is the remainder when 113 is divided by 9? (a) 5 (b) 7 (c) 6 (d) 4
23. 23.What is the remainder when 264 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
24. 24.What is the smallest number you must add to 61 to make it divisible by 3? (a) 2 (b) 1
25. 25.What is the smallest number you must add to 144 to make it divisible by 10? (a) 7 (b) 6 (c) 4 (d) 5
26. 26.What is the remainder when 45 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
27. 27.Which of these numbers is divisible by 3? (a) 1910 (b) 1906 (c) 1913 (d) 1908
28. 28.What is the remainder when 138 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
29. 29.Which of these numbers is divisible by 8? (a) 5890 (b) 5886 (c) 5888 (d) 5897
30. 30.What is the units digit of 4^16? (a) 5 (b) 7 (c) 6 (d) 4
31. 31.What is the units digit of 8^41? (a) 2 (b) 6 (c) 8 (d) 4
32. 32.What is the units digit of 3^14? (a) 7 (b) 3 (c) 9 (d) 1
33. 33.Which of these numbers is divisible by 3? (a) 1933 (b) 1928 (c) 1930 (d) 1929
34. 34.What is the smallest number you must add to 134 to make it divisible by 4? (a) 3 (b) 2 (c) 1
35. 35.Which of these numbers is divisible by 6? (a) 5237 (b) 5238 (c) 5241 (d) 5246
36. 36.What is the units digit of 2^31? (a) 8 (b) 2 (c) 6 (d) 4
37. 37.Which of these numbers is divisible by 3? (a) 120 (b) 118 (c) 124 (d) 122
38. 38.What is the remainder when 530 is divided by 8? (a) 3 (b) 1 (c) 6 (d) 2
39. 39.What is the remainder when 337 is divided by 9? (a) 3 (b) 4 (c) 5 (d) 6
40. 40.What is the smallest number you must add to 191 to make it divisible by 6? (a) 1 (b) 5 (c) 2 (d) 3

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