Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 387 · 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 3? (a) 2213 (b) 2214 (c) 2215 (d) 2219
2. 2.What is the remainder when 234 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
3. 3.What is the units digit of 3^45? (a) 1 (b) 3 (c) 7 (d) 9
4. 4.What is the smallest number you must add to 334 to make it divisible by 9? (a) 7 (b) 8 (c) 6 (d) 1
5. 5.What is the units digit of 9^51? (a) 8 (b) 7 (c) 9 (d) 1
6. 6.What is the units digit of 8^50? (a) 2 (b) 4 (c) 8 (d) 6
7. 7.Which of these numbers is divisible by 3? (a) 2473 (b) 2471 (c) 2472 (d) 2477
8. 8.Which of these numbers is divisible by 3? (a) 1556 (b) 1561 (c) 1558 (d) 1557
9. 9.Which of these numbers is divisible by 3? (a) 748 (b) 752 (c) 745 (d) 747
10. 10.What is the units digit of 2^29? (a) 6 (b) 4 (c) 8 (d) 2
11. 11.Which of these numbers is divisible by 6? (a) 2195 (b) 2200 (c) 2196 (d) 2207
12. 12.Which of these numbers is divisible by 9? (a) 3208 (b) 3203 (c) 3217 (d) 3204
13. 13.Which of these numbers is divisible by 3? (a) 1926 (b) 1924 (c) 1928 (d) 1930
14. 14.Which of these numbers is divisible by 4? (a) 3177 (b) 3182 (c) 3185 (d) 3180
15. 15.What is the units digit of 2^18? (a) 2 (b) 8 (c) 6 (d) 4
16. 16.What is the units digit of 7^34? (a) 1 (b) 3 (c) 7 (d) 9
17. 17.What is the remainder when 111 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
18. 18.What is the remainder when 379 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
19. 19.Which of these numbers is divisible by 8? (a) 6409 (b) 6408 (c) 6407 (d) 6420
20. 20.What is the remainder when 163 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
21. 21.What is the units digit of 4^49? (a) 4 (b) 6 (c) 3 (d) 5
22. 22.Which of these numbers is divisible by 3? (a) 694 (b) 692 (c) 693 (d) 698
23. 23.Which of these numbers is divisible by 8? (a) 842 (b) 832 (c) 831 (d) 839
24. 24.Which of these numbers is divisible by 8? (a) 6236 (b) 6226 (c) 6242 (d) 6232
25. 25.What is the remainder when 222 is divided by 8? (a) 2 (b) 5 (c) 7 (d) 6
26. 26.What is the remainder when 722 is divided by 9? (a) 3 (b) 1 (c) 7 (d) 2
27. 27.What is the smallest number you must add to 571 to make it divisible by 8? (a) 6 (b) 5 (c) 3 (d) 4
28. 28.What is the smallest number you must add to 279 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
29. 29.Which of these numbers is divisible by 9? (a) 3689 (b) 3675 (c) 3667 (d) 3672
30. 30.Which of these numbers is divisible by 4? (a) 2609 (b) 2602 (c) 2605 (d) 2604
31. 31.What is the remainder when 143 is divided by 3? (a) 1 (b) 0 (c) 2
32. 32.What is the smallest number you must add to 282 to make it divisible by 4? (a) 3 (b) 1 (c) 2
33. 33.What is the remainder when 263 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
34. 34.Which of these numbers is divisible by 8? (a) 2161 (b) 2152 (c) 2147 (d) 2153
35. 35.What is the remainder when 82 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 2
36. 36.Which of these numbers is divisible by 9? (a) 1405 (b) 1397 (c) 1416 (d) 1404
37. 37.What is the remainder when 829 is divided by 12? (a) 2 (b) 1 (c) 0 (d) 11
38. 38.What is the units digit of 7^36? (a) 3 (b) 9 (c) 1 (d) 7
39. 39.Which of these numbers is divisible by 8? (a) 1159 (b) 1170 (c) 1162 (d) 1160
40. 40.What is the remainder when 254 is divided by 11? (a) 0 (b) 10 (c) 1 (d) 2

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