Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 458 · 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 133 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
2. 2.Which of these numbers is divisible by 11? (a) 8923 (b) 8903 (c) 8910 (d) 8914
3. 3.What is the units digit of 2^21? (a) 2 (b) 6 (c) 8 (d) 4
4. 4.Which of these numbers is divisible by 6? (a) 984 (b) 988 (c) 992 (d) 982
5. 5.What is the smallest number you must add to 287 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
6. 6.What is the remainder when 564 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
7. 7.Which of these numbers is divisible by 11? (a) 3087 (b) 3074 (c) 3080 (d) 3101
8. 8.Which of these numbers is divisible by 3? (a) 1442 (b) 1448 (c) 1444 (d) 1443
9. 9.What is the remainder when 452 is divided by 9? (a) 2 (b) 3 (c) 7 (d) 1
10. 10.What is the smallest number you must add to 109 to make it divisible by 6? (a) 5 (b) 4 (c) 1 (d) 3
11. 11.What is the smallest number you must add to 381 to make it divisible by 9? (a) 7 (b) 6 (c) 5 (d) 3
12. 12.What is the smallest number you must add to 265 to make it divisible by 4? (a) 1 (b) 3 (c) 2
13. 13.What is the smallest number you must add to 375 to make it divisible by 10? (a) 5 (b) 4 (c) 6 (d) 7
14. 14.What is the units digit of 3^48? (a) 1 (b) 7 (c) 3 (d) 9
15. 15.Which of these numbers is divisible by 3? (a) 2039 (b) 2041 (c) 2044 (d) 2040
16. 16.Which of these numbers is divisible by 6? (a) 1358 (b) 1365 (c) 1372 (d) 1362
17. 17.What is the remainder when 179 is divided by 3? (a) 1 (b) 0 (c) 2
18. 18.Which of these numbers is divisible by 9? (a) 4221 (b) 4218 (c) 4225 (d) 4235
19. 19.What is the remainder when 387 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
20. 20.Which of these numbers is divisible by 9? (a) 7537 (b) 7526 (c) 7533 (d) 7546
21. 21.Which of these numbers is divisible by 9? (a) 494 (b) 479 (c) 477 (d) 473
22. 22.Which of these numbers is divisible by 6? (a) 263 (b) 266 (c) 272 (d) 264
23. 23.What is the smallest number you must add to 467 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
24. 24.What is the smallest number you must add to 198 to make it divisible by 8? (a) 3 (b) 6 (c) 1 (d) 2
25. 25.What is the remainder when 430 is divided by 8? (a) 5 (b) 6 (c) 2 (d) 7
26. 26.Which of these numbers is divisible by 8? (a) 1954 (b) 1971 (c) 1967 (d) 1960
27. 27.What is the units digit of 8^30? (a) 2 (b) 6 (c) 4 (d) 8
28. 28.What is the units digit of 2^46? (a) 8 (b) 6 (c) 4 (d) 2
29. 29.Which of these numbers is divisible by 9? (a) 6369 (b) 6372 (c) 6384 (d) 6374
30. 30.Which of these numbers is divisible by 3? (a) 2024 (b) 2029 (c) 2025 (d) 2026
31. 31.What is the smallest number you must add to 437 to make it divisible by 9? (a) 5 (b) 4 (c) 6 (d) 3
32. 32.What is the units digit of 8^43? (a) 4 (b) 6 (c) 8 (d) 2
33. 33.What is the units digit of 9^43? (a) 8 (b) 7 (c) 9 (d) 1
34. 34.Which of these numbers is divisible by 4? (a) 199 (b) 202 (c) 200 (d) 205
35. 35.Which of these numbers is divisible by 3? (a) 1275 (b) 1280 (c) 1277 (d) 1274
36. 36.Which of these numbers is divisible by 3? (a) 560 (b) 563 (c) 565 (d) 561
37. 37.What is the units digit of 7^14? (a) 9 (b) 3 (c) 7 (d) 1
38. 38.What is the remainder when 328 is divided by 10? (a) 7 (b) 9 (c) 2 (d) 8
39. 39.Which of these numbers is divisible by 3? (a) 122 (b) 124 (c) 120 (d) 118
40. 40.Which of these numbers is divisible by 11? (a) 6577 (b) 6581 (c) 6566 (d) 6567

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