Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 12 · 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 371 to make it divisible by 8? (a) 5 (b) 4 (c) 6 (d) 3
2. 2.Which of these numbers is divisible by 3? (a) 2019 (b) 2021 (c) 2023 (d) 2018
3. 3.Which of these numbers is divisible by 6? (a) 5250 (b) 5251 (c) 5248 (d) 5257
4. 4.Which of these numbers is divisible by 8? (a) 6733 (b) 6726 (c) 6737 (d) 6728
5. 5.What is the smallest number you must add to 93 to make it divisible by 4? (a) 1 (b) 2 (c) 3
6. 6.Which of these numbers is divisible by 8? (a) 1266 (b) 1259 (c) 1256 (d) 1254
7. 7.What is the units digit of 7^31? (a) 3 (b) 7 (c) 9 (d) 1
8. 8.What is the units digit of 4^53? (a) 3 (b) 5 (c) 4 (d) 6
9. 9.What is the smallest number you must add to 99 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 1
10. 10.Which of these numbers is divisible by 4? (a) 1826 (b) 1822 (c) 1820 (d) 1817
11. 11.What is the units digit of 4^59? (a) 6 (b) 5 (c) 4 (d) 3
12. 12.Which of these numbers is divisible by 4? (a) 1877 (b) 1874 (c) 1871 (d) 1872
13. 13.What is the units digit of 7^17? (a) 9 (b) 3 (c) 1 (d) 7
14. 14.What is the smallest number you must add to 793 to make it divisible by 12? (a) 1 (b) 9 (c) 11 (d) 10
15. 15.What is the remainder when 587 is divided by 10? (a) 3 (b) 6 (c) 7 (d) 8
16. 16.Which of these numbers is divisible by 8? (a) 918 (b) 931 (c) 921 (d) 920
17. 17.What is the smallest number you must add to 703 to make it divisible by 9? (a) 1 (b) 7 (c) 8 (d) 6
18. 18.What is the remainder when 772 is divided by 10? (a) 2 (b) 3 (c) 8 (d) 1
19. 19.What is the smallest number you must add to 247 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 6
20. 20.What is the smallest number you must add to 458 to make it divisible by 7? (a) 5 (b) 6 (c) 3 (d) 4
21. 21.Which of these numbers is divisible by 4? (a) 1379 (b) 1381 (c) 1380 (d) 1387
22. 22.What is the units digit of 3^29? (a) 3 (b) 7 (c) 9 (d) 1
23. 23.What is the smallest number you must add to 239 to make it divisible by 4? (a) 3 (b) 1 (c) 2
24. 24.What is the units digit of 8^25? (a) 4 (b) 6 (c) 2 (d) 8
25. 25.Which of these numbers is divisible by 3? (a) 300 (b) 299 (c) 304 (d) 301
26. 26.What is the remainder when 77 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
27. 27.What is the units digit of 9^35? (a) 7 (b) 8 (c) 9 (d) 1
28. 28.Which of these numbers is divisible by 8? (a) 1849 (b) 1857 (c) 1848 (d) 1841
29. 29.What is the smallest number you must add to 176 to make it divisible by 12? (a) 5 (b) 8 (c) 3 (d) 4
30. 30.What is the remainder when 281 is divided by 7? (a) 0 (b) 1 (c) 2 (d) 6
31. 31.What is the units digit of 8^39? (a) 2 (b) 8 (c) 6 (d) 4
32. 32.Which of these numbers is divisible by 6? (a) 5281 (b) 5280 (c) 5276 (d) 5291
33. 33.Which of these numbers is divisible by 11? (a) 3100 (b) 3120 (c) 3102 (d) 3103
34. 34.What is the remainder when 752 is divided by 11? (a) 4 (b) 5 (c) 3 (d) 7
35. 35.What is the remainder when 308 is divided by 12? (a) 8 (b) 4 (c) 9 (d) 7
36. 36.What is the smallest number you must add to 88 to make it divisible by 6? (a) 2 (b) 4 (c) 1 (d) 3
37. 37.What is the smallest number you must add to 608 to make it divisible by 12? (a) 3 (b) 5 (c) 4 (d) 8
38. 38.Which of these numbers is divisible by 3? (a) 1968 (b) 1966 (c) 1972 (d) 1969
39. 39.What is the remainder when 384 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
40. 40.What is the smallest number you must add to 79 to make it divisible by 3? (a) 2 (b) 1

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