Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 19 · 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) 1279 (b) 1280 (c) 1281 (d) 1287
2. 2.Which of these numbers is divisible by 4? (a) 1023 (b) 1025 (c) 1029 (d) 1024
3. 3.What is the smallest number you must add to 383 to make it divisible by 9? (a) 5 (b) 6 (c) 3 (d) 4
4. 4.Which of these numbers is divisible by 9? (a) 7990 (b) 7996 (c) 7982 (d) 7983
5. 5.Which of these numbers is divisible by 9? (a) 2206 (b) 2198 (c) 2205 (d) 2215
6. 6.What is the remainder when 381 is divided by 10? (a) 9 (b) 1 (c) 2 (d) 0
7. 7.What is the units digit of 3^57? (a) 3 (b) 7 (c) 1 (d) 9
8. 8.What is the units digit of 2^25? (a) 6 (b) 8 (c) 2 (d) 4
9. 9.Which of these numbers is divisible by 9? (a) 744 (b) 754 (c) 738 (d) 733
10. 10.What is the units digit of 4^58? (a) 5 (b) 4 (c) 6 (d) 7
11. 11.What is the smallest number you must add to 468 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 6
12. 12.What is the smallest number you must add to 248 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
13. 13.What is the units digit of 8^56? (a) 8 (b) 4 (c) 6 (d) 2
14. 14.What is the remainder when 237 is divided by 9? (a) 3 (b) 4 (c) 6 (d) 2
15. 15.What is the units digit of 3^15? (a) 7 (b) 3 (c) 1 (d) 9
16. 16.Which of these numbers is divisible by 6? (a) 2564 (b) 2556 (c) 2560 (d) 2553
17. 17.What is the remainder when 403 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
18. 18.What is the remainder when 183 is divided by 10? (a) 7 (b) 4 (c) 2 (d) 3
19. 19.What is the units digit of 9^31? (a) 9 (b) 1 (c) 7 (d) 8
20. 20.What is the units digit of 8^21? (a) 2 (b) 4 (c) 8 (d) 6
21. 21.What is the remainder when 280 is divided by 12? (a) 3 (b) 8 (c) 4 (d) 5
22. 22.Which of these numbers is divisible by 4? (a) 1203 (b) 1196 (c) 1193 (d) 1198
23. 23.What is the remainder when 195 is divided by 7? (a) 1 (b) 5 (c) 4 (d) 6
24. 24.Which of these numbers is divisible by 11? (a) 4617 (b) 4625 (c) 4632 (d) 4620
25. 25.What is the remainder when 252 is divided by 10? (a) 1 (b) 8 (c) 3 (d) 2
26. 26.What is the units digit of 4^30? (a) 6 (b) 5 (c) 7 (d) 4
27. 27.Which of these numbers is divisible by 3? (a) 1940 (b) 1937 (c) 1942 (d) 1938
28. 28.Which of these numbers is divisible by 3? (a) 1385 (b) 1388 (c) 1381 (d) 1383
29. 29.What is the remainder when 824 is divided by 11? (a) 1 (b) 9 (c) 8 (d) 10
30. 30.What is the units digit of 7^39? (a) 7 (b) 9 (c) 1 (d) 3
31. 31.What is the units digit of 8^33? (a) 2 (b) 6 (c) 8 (d) 4
32. 32.What is the remainder when 459 is divided by 12? (a) 4 (b) 3 (c) 9 (d) 2
33. 33.What is the units digit of 7^48? (a) 9 (b) 7 (c) 3 (d) 1
34. 34.What is the smallest number you must add to 412 to make it divisible by 11? (a) 7 (b) 8 (c) 5 (d) 6
35. 35.What is the units digit of 9^57? (a) 8 (b) 7 (c) 1 (d) 9
36. 36.What is the units digit of 3^35? (a) 9 (b) 7 (c) 1 (d) 3
37. 37.Which of these numbers is divisible by 8? (a) 6719 (b) 6706 (c) 6712 (d) 6724
38. 38.Which of these numbers is divisible by 3? (a) 299 (b) 294 (c) 296 (d) 293
39. 39.What is the units digit of 8^47? (a) 6 (b) 8 (c) 4 (d) 2
40. 40.What is the smallest number you must add to 144 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1

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