Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 25 · 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 9? (a) 1920 (b) 1931 (c) 1914 (d) 1917
2. 2.What is the units digit of 4^25? (a) 3 (b) 5 (c) 6 (d) 4
3. 3.Which of these numbers is divisible by 6? (a) 2538 (b) 2535 (c) 2540 (d) 2547
4. 4.What is the units digit of 2^17? (a) 2 (b) 6 (c) 8 (d) 4
5. 5.Which of these numbers is divisible by 3? (a) 301 (b) 299 (c) 295 (d) 297
6. 6.What is the remainder when 184 is divided by 10? (a) 4 (b) 3 (c) 5 (d) 6
7. 7.Which of these numbers is divisible by 9? (a) 3605 (b) 3597 (c) 3600 (d) 3610
8. 8.What is the smallest number you must add to 277 to make it divisible by 4? (a) 2 (b) 3 (c) 1
9. 9.What is the units digit of 7^55? (a) 9 (b) 1 (c) 7 (d) 3
10. 10.What is the units digit of 2^26? (a) 8 (b) 2 (c) 6 (d) 4
11. 11.Which of these numbers is divisible by 8? (a) 5830 (b) 5824 (c) 5821 (d) 5835
12. 12.What is the smallest number you must add to 80 to make it divisible by 3? (a) 2 (b) 1
13. 13.What is the remainder when 574 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 3
14. 14.Which of these numbers is divisible by 3? (a) 1421 (b) 1418 (c) 1424 (d) 1419
15. 15.What is the smallest number you must add to 501 to make it divisible by 7? (a) 4 (b) 3 (c) 5 (d) 2
16. 16.What is the units digit of 3^50? (a) 3 (b) 7 (c) 9 (d) 1
17. 17.What is the smallest number you must add to 628 to make it divisible by 9? (a) 1 (b) 7 (c) 3 (d) 2
18. 18.Which of these numbers is divisible by 11? (a) 4810 (b) 4799 (c) 4787 (d) 4796
19. 19.Which of these numbers is divisible by 11? (a) 3258 (b) 3245 (c) 3242 (d) 3246
20. 20.What is the remainder when 348 is divided by 7? (a) 6 (b) 2 (c) 4 (d) 5
21. 21.What is the units digit of 4^53? (a) 3 (b) 5 (c) 4 (d) 6
22. 22.What is the smallest number you must add to 313 to make it divisible by 4? (a) 2 (b) 1 (c) 3
23. 23.What is the units digit of 7^15? (a) 7 (b) 3 (c) 9 (d) 1
24. 24.Which of these numbers is divisible by 11? (a) 3417 (b) 3398 (c) 3399 (d) 3406
25. 25.What is the remainder when 428 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
26. 26.What is the remainder when 130 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 2
27. 27.What is the remainder when 177 is divided by 12? (a) 3 (b) 10 (c) 9 (d) 8
28. 28.What is the smallest number you must add to 461 to make it divisible by 8? (a) 2 (b) 5 (c) 3 (d) 4
29. 29.Which of these numbers is divisible by 11? (a) 3801 (b) 3807 (c) 3806 (d) 3825
30. 30.What is the remainder when 190 is divided by 3? (a) 2 (b) 1 (c) 0
31. 31.Which of these numbers is divisible by 11? (a) 8186 (b) 8202 (c) 8175 (d) 8184
32. 32.Which of these numbers is divisible by 11? (a) 6372 (b) 6387 (c) 6380 (d) 6400
33. 33.Which of these numbers is divisible by 8? (a) 2202 (b) 2194 (c) 2192 (d) 2186
34. 34.What is the units digit of 4^10? (a) 5 (b) 4 (c) 6 (d) 7
35. 35.Which of these numbers is divisible by 3? (a) 1183 (b) 1190 (c) 1185 (d) 1187
36. 36.Which of these numbers is divisible by 4? (a) 2439 (b) 2440 (c) 2441 (d) 2445
37. 37.What is the units digit of 9^37? (a) 9 (b) 8 (c) 1 (d) 7
38. 38.What is the smallest number you must add to 95 to make it divisible by 4? (a) 3 (b) 2 (c) 1
39. 39.Which of these numbers is divisible by 6? (a) 1594 (b) 1596 (c) 1600 (d) 1605
40. 40.What is the remainder when 351 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4

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