Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 370 · 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 11? (a) 1861 (b) 1851 (c) 1880 (d) 1859
2. 2.What is the smallest number you must add to 545 to make it divisible by 9? (a) 6 (b) 4 (c) 5 (d) 3
3. 3.What is the smallest number you must add to 569 to make it divisible by 12? (a) 6 (b) 5 (c) 7 (d) 8
4. 4.What is the remainder when 463 is divided by 9? (a) 5 (b) 6 (c) 4 (d) 3
5. 5.What is the smallest number you must add to 358 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
6. 6.What is the units digit of 2^38? (a) 4 (b) 6 (c) 8 (d) 2
7. 7.What is the remainder when 756 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
8. 8.What is the remainder when 170 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
9. 9.Which of these numbers is divisible by 3? (a) 2044 (b) 2046 (c) 2051 (d) 2047
10. 10.What is the units digit of 2^19? (a) 2 (b) 6 (c) 8 (d) 4
11. 11.What is the remainder when 215 is divided by 3? (a) 1 (b) 0 (c) 2
12. 12.What is the smallest number you must add to 340 to make it divisible by 11? (a) 1 (b) 3 (c) 2 (d) 10
13. 13.What is the remainder when 480 is divided by 9? (a) 4 (b) 2 (c) 6 (d) 3
14. 14.Which of these numbers is divisible by 8? (a) 6560 (b) 6571 (c) 6556 (d) 6561
15. 15.What is the units digit of 4^15? (a) 6 (b) 4 (c) 5 (d) 3
16. 16.What is the smallest number you must add to 149 to make it divisible by 6? (a) 1 (b) 2 (c) 3 (d) 5
17. 17.What is the smallest number you must add to 368 to make it divisible by 9? (a) 2 (b) 3 (c) 8 (d) 1
18. 18.What is the remainder when 168 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
19. 19.What is the units digit of 7^44? (a) 3 (b) 1 (c) 9 (d) 7
20. 20.What is the smallest number you must add to 122 to make it divisible by 3? (a) 1 (b) 2
21. 21.What is the smallest number you must add to 618 to make it divisible by 8? (a) 7 (b) 5 (c) 6 (d) 2
22. 22.Which of these numbers is divisible by 9? (a) 3501 (b) 3513 (c) 3506 (d) 3496
23. 23.What is the remainder when 417 is divided by 10? (a) 3 (b) 6 (c) 8 (d) 7
24. 24.What is the remainder when 193 is divided by 7? (a) 3 (b) 6 (c) 5 (d) 4
25. 25.What is the smallest number you must add to 228 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
26. 26.What is the smallest number you must add to 209 to make it divisible by 4? (a) 1 (b) 2 (c) 3
27. 27.What is the units digit of 7^29? (a) 7 (b) 3 (c) 9 (d) 1
28. 28.Which of these numbers is divisible by 11? (a) 7242 (b) 7228 (c) 7238 (d) 7259
29. 29.What is the remainder when 132 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
30. 30.What is the units digit of 7^33? (a) 7 (b) 3 (c) 1 (d) 9
31. 31.What is the remainder when 43 is divided by 3? (a) 2 (b) 0 (c) 1
32. 32.Which of these numbers is divisible by 11? (a) 921 (b) 902 (c) 897 (d) 904
33. 33.Which of these numbers is divisible by 6? (a) 2604 (b) 2605 (c) 2615 (d) 2602
34. 34.What is the smallest number you must add to 250 to make it divisible by 6? (a) 2 (b) 1 (c) 4 (d) 3
35. 35.What is the remainder when 128 is divided by 6? (a) 2 (b) 1 (c) 3 (d) 4
36. 36.What is the smallest number you must add to 622 to make it divisible by 12? (a) 1 (b) 3 (c) 2 (d) 10
37. 37.What is the units digit of 8^22? (a) 8 (b) 6 (c) 2 (d) 4
38. 38.What is the smallest number you must add to 645 to make it divisible by 9? (a) 6 (b) 2 (c) 4 (d) 3
39. 39.What is the remainder when 256 is divided by 10? (a) 6 (b) 4 (c) 7 (d) 5
40. 40.What is the remainder when 206 is divided by 3? (a) 0 (b) 1 (c) 2

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