Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 388 · 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 units digit of 9^51? (a) 1 (b) 7 (c) 9 (d) 8
2. 2.Which of these numbers is divisible by 3? (a) 1479 (b) 1477 (c) 1483 (d) 1480
3. 3.What is the units digit of 7^27? (a) 7 (b) 9 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 3? (a) 2425 (b) 2424 (c) 2423 (d) 2429
5. 5.What is the units digit of 2^31? (a) 4 (b) 2 (c) 6 (d) 8
6. 6.What is the units digit of 8^22? (a) 6 (b) 4 (c) 2 (d) 8
7. 7.Which of these numbers is divisible by 11? (a) 5890 (b) 5902 (c) 5914 (d) 5896
8. 8.What is the smallest number you must add to 410 to make it divisible by 8? (a) 7 (b) 6 (c) 2 (d) 5
9. 9.What is the remainder when 64 is divided by 3? (a) 1 (b) 0 (c) 2
10. 10.What is the remainder when 246 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
11. 11.What is the smallest number you must add to 789 to make it divisible by 11? (a) 3 (b) 2 (c) 8 (d) 4
12. 12.Which of these numbers is divisible by 11? (a) 2260 (b) 2249 (c) 2255 (d) 2274
13. 13.Which of these numbers is divisible by 4? (a) 657 (b) 662 (c) 666 (d) 660
14. 14.What is the remainder when 82 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
15. 15.What is the units digit of 3^26? (a) 7 (b) 9 (c) 3 (d) 1
16. 16.What is the smallest number you must add to 115 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 4
17. 17.What is the smallest number you must add to 107 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
18. 18.What is the remainder when 229 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
19. 19.Which of these numbers is divisible by 4? (a) 3423 (b) 3419 (c) 3420 (d) 3425
20. 20.What is the remainder when 493 is divided by 9? (a) 6 (b) 8 (c) 2 (d) 7
21. 21.What is the remainder when 204 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
22. 22.Which of these numbers is divisible by 9? (a) 544 (b) 561 (c) 549 (d) 557
23. 23.What is the units digit of 2^12? (a) 8 (b) 6 (c) 4 (d) 2
24. 24.What is the units digit of 2^37? (a) 4 (b) 8 (c) 6 (d) 2
25. 25.Which of these numbers is divisible by 3? (a) 529 (b) 533 (c) 528 (d) 527
26. 26.What is the smallest number you must add to 83 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
27. 27.What is the remainder when 122 is divided by 6? (a) 4 (b) 2 (c) 1 (d) 3
28. 28.Which of these numbers is divisible by 6? (a) 2767 (b) 2764 (c) 2766 (d) 2776
29. 29.Which of these numbers is divisible by 8? (a) 5753 (b) 5762 (c) 5751 (d) 5752
30. 30.What is the smallest number you must add to 968 to make it divisible by 12? (a) 4 (b) 5 (c) 3 (d) 8
31. 31.Which of these numbers is divisible by 3? (a) 2157 (b) 2161 (c) 2156 (d) 2159
32. 32.Which of these numbers is divisible by 9? (a) 7551 (b) 7565 (c) 7545 (d) 7559
33. 33.What is the units digit of 4^43? (a) 6 (b) 5 (c) 4 (d) 3
34. 34.What is the smallest number you must add to 326 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
35. 35.What is the smallest number you must add to 186 to make it divisible by 7? (a) 4 (b) 3 (c) 2 (d) 5
36. 36.Which of these numbers is divisible by 4? (a) 225 (b) 222 (c) 218 (d) 220
37. 37.Which of these numbers is divisible by 4? (a) 2954 (b) 2952 (c) 2951 (d) 2959
38. 38.What is the units digit of 4^21? (a) 4 (b) 6 (c) 3 (d) 5
39. 39.What is the smallest number you must add to 643 to make it divisible by 12? (a) 4 (b) 6 (c) 5 (d) 7
40. 40.What is the units digit of 9^31? (a) 1 (b) 9 (c) 7 (d) 8

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