Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 292 · 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 283 to make it divisible by 4? (a) 2 (b) 1 (c) 3
2. 2.Which of these numbers is divisible by 3? (a) 255 (b) 254 (c) 259 (d) 256
3. 3.Which of these numbers is divisible by 8? (a) 6287 (b) 6293 (c) 6277 (d) 6280
4. 4.Which of these numbers is divisible by 4? (a) 2180 (b) 2177 (c) 2186 (d) 2181
5. 5.What is the units digit of 7^10? (a) 1 (b) 3 (c) 7 (d) 9
6. 6.What is the smallest number you must add to 327 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 5
7. 7.What is the units digit of 2^48? (a) 4 (b) 6 (c) 2 (d) 8
8. 8.What is the smallest number you must add to 638 to make it divisible by 8? (a) 1 (b) 6 (c) 2 (d) 3
9. 9.What is the smallest number you must add to 357 to make it divisible by 9? (a) 4 (b) 3 (c) 2 (d) 6
10. 10.Which of these numbers is divisible by 3? (a) 1980 (b) 1981 (c) 1979 (d) 1985
11. 11.Which of these numbers is divisible by 8? (a) 1384 (b) 1391 (c) 1397 (d) 1380
12. 12.Which of these numbers is divisible by 6? (a) 5084 (b) 5091 (c) 5081 (d) 5082
13. 13.What is the units digit of 7^48? (a) 9 (b) 7 (c) 1 (d) 3
14. 14.Which of these numbers is divisible by 8? (a) 503 (b) 518 (c) 510 (d) 504
15. 15.What is the remainder when 132 is divided by 8? (a) 4 (b) 6 (c) 3 (d) 5
16. 16.Which of these numbers is divisible by 4? (a) 593 (b) 599 (c) 592 (d) 591
17. 17.What is the remainder when 207 is divided by 8? (a) 7 (b) 6 (c) 5 (d) 1
18. 18.Which of these numbers is divisible by 11? (a) 729 (b) 744 (c) 737 (d) 750
19. 19.What is the smallest number you must add to 199 to make it divisible by 8? (a) 3 (b) 2 (c) 1 (d) 7
20. 20.What is the remainder when 430 is divided by 9? (a) 6 (b) 2 (c) 7 (d) 8
21. 21.What is the remainder when 70 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 2
22. 22.Which of these numbers is divisible by 11? (a) 8660 (b) 8668 (c) 8681 (d) 8677
23. 23.What is the units digit of 3^10? (a) 3 (b) 7 (c) 9 (d) 1
24. 24.What is the units digit of 4^34? (a) 6 (b) 7 (c) 4 (d) 5
25. 25.Which of these numbers is divisible by 4? (a) 2774 (b) 2772 (c) 2779 (d) 2769
26. 26.What is the smallest number you must add to 124 to make it divisible by 3? (a) 1 (b) 2
27. 27.Which of these numbers is divisible by 4? (a) 3460 (b) 3466 (c) 3462 (d) 3458
28. 28.Which of these numbers is divisible by 3? (a) 1385 (b) 1379 (c) 1382 (d) 1380
29. 29.What is the smallest number you must add to 92 to make it divisible by 7? (a) 1 (b) 6 (c) 5 (d) 4
30. 30.What is the units digit of 2^34? (a) 2 (b) 4 (c) 8 (d) 6
31. 31.What is the units digit of 4^57? (a) 4 (b) 3 (c) 5 (d) 6
32. 32.What is the smallest number you must add to 349 to make it divisible by 6? (a) 3 (b) 1 (c) 5 (d) 4
33. 33.What is the remainder when 238 is divided by 8? (a) 7 (b) 5 (c) 6 (d) 2
34. 34.Which of these numbers is divisible by 11? (a) 1158 (b) 1172 (c) 1155 (d) 1153
35. 35.Which of these numbers is divisible by 11? (a) 5084 (b) 5098 (c) 5082 (d) 5076
36. 36.What is the units digit of 2^25? (a) 4 (b) 6 (c) 2 (d) 8
37. 37.What is the units digit of 9^31? (a) 7 (b) 8 (c) 9 (d) 1
38. 38.Which of these numbers is divisible by 4? (a) 334 (b) 341 (c) 336 (d) 337
39. 39.Which of these numbers is divisible by 8? (a) 5666 (b) 5674 (c) 5664 (d) 5663
40. 40.Which of these numbers is divisible by 3? (a) 473 (b) 471 (c) 470 (d) 475

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