Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 836 · 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) 1444 (b) 1438 (c) 1442 (d) 1440
2. 2.What is the remainder when 289 is divided by 9? (a) 0 (b) 8 (c) 2 (d) 1
3. 3.Which of these numbers is divisible by 4? (a) 2300 (b) 2307 (c) 2303 (d) 2298
4. 4.What is the units digit of 9^15? (a) 7 (b) 8 (c) 9 (d) 1
5. 5.Which of these numbers is divisible by 9? (a) 3211 (b) 3216 (c) 3224 (d) 3213
6. 6.What is the smallest number you must add to 324 to make it divisible by 11? (a) 8 (b) 6 (c) 5 (d) 7
7. 7.What is the smallest number you must add to 143 to make it divisible by 6? (a) 2 (b) 1 (c) 5 (d) 3
8. 8.What is the remainder when 518 is divided by 12? (a) 2 (b) 3 (c) 1 (d) 10
9. 9.Which of these numbers is divisible by 11? (a) 2477 (b) 2475 (c) 2471 (d) 2488
10. 10.What is the units digit of 4^26? (a) 4 (b) 7 (c) 6 (d) 5
11. 11.Which of these numbers is divisible by 8? (a) 353 (b) 349 (c) 342 (d) 344
12. 12.What is the units digit of 9^16? (a) 2 (b) 9 (c) 0 (d) 1
13. 13.What is the smallest number you must add to 444 to make it divisible by 9? (a) 5 (b) 3 (c) 6 (d) 7
14. 14.What is the units digit of 9^12? (a) 0 (b) 1 (c) 2 (d) 9
15. 15.Which of these numbers is divisible by 11? (a) 6320 (b) 6293 (c) 6303 (d) 6311
16. 16.Which of these numbers is divisible by 6? (a) 3686 (b) 3693 (c) 3699 (d) 3690
17. 17.What is the units digit of 8^37? (a) 2 (b) 8 (c) 6 (d) 4
18. 18.What is the remainder when 67 is divided by 6? (a) 2 (b) 5 (c) 1 (d) 0
19. 19.What is the remainder when 32 is divided by 3? (a) 0 (b) 2 (c) 1
20. 20.What is the units digit of 4^20? (a) 6 (b) 4 (c) 5 (d) 7
21. 21.What is the units digit of 2^52? (a) 8 (b) 2 (c) 6 (d) 4
22. 22.Which of these numbers is divisible by 9? (a) 5722 (b) 5724 (c) 5732 (d) 5736
23. 23.What is the remainder when 466 is divided by 8? (a) 3 (b) 1 (c) 2 (d) 6
24. 24.What is the smallest number you must add to 174 to make it divisible by 9? (a) 7 (b) 5 (c) 6 (d) 3
25. 25.What is the remainder when 734 is divided by 12? (a) 2 (b) 3 (c) 10 (d) 1
26. 26.Which of these numbers is divisible by 4? (a) 873 (b) 876 (c) 881 (d) 878
27. 27.What is the remainder when 190 is divided by 6? (a) 5 (b) 3 (c) 2 (d) 4
28. 28.Which of these numbers is divisible by 4? (a) 1979 (b) 1980 (c) 1985 (d) 1982
29. 29.Which of these numbers is divisible by 4? (a) 261 (b) 267 (c) 259 (d) 260
30. 30.What is the smallest number you must add to 708 to make it divisible by 11? (a) 6 (b) 4 (c) 7 (d) 8
31. 31.Which of these numbers is divisible by 8? (a) 6529 (b) 6547 (c) 6536 (d) 6540
32. 32.What is the units digit of 7^30? (a) 7 (b) 3 (c) 1 (d) 9
33. 33.What is the smallest number you must add to 331 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
34. 34.What is the units digit of 3^52? (a) 1 (b) 9 (c) 3 (d) 7
35. 35.Which of these numbers is divisible by 3? (a) 647 (b) 649 (c) 652 (d) 648
36. 36.What is the units digit of 2^57? (a) 4 (b) 8 (c) 2 (d) 6
37. 37.Which of these numbers is divisible by 6? (a) 4597 (b) 4604 (c) 4611 (d) 4602
38. 38.What is the smallest number you must add to 81 to make it divisible by 7? (a) 2 (b) 3 (c) 4 (d) 5
39. 39.What is the units digit of 9^14? (a) 2 (b) 0 (c) 1 (d) 9
40. 40.Which of these numbers is divisible by 9? (a) 2382 (b) 2373 (c) 2367 (d) 2366

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