Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 703 · 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) 2189 (b) 2187 (c) 2201 (d) 2190
2. 2.Which of these numbers is divisible by 3? (a) 1925 (b) 1922 (c) 1919 (d) 1920
3. 3.Which of these numbers is divisible by 11? (a) 3309 (b) 3320 (c) 3331 (d) 3311
4. 4.What is the units digit of 8^28? (a) 2 (b) 4 (c) 8 (d) 6
5. 5.Which of these numbers is divisible by 9? (a) 3818 (b) 3825 (c) 3828 (d) 3837
6. 6.Which of these numbers is divisible by 6? (a) 2906 (b) 2910 (c) 2913 (d) 2917
7. 7.What is the smallest number you must add to 251 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
8. 8.What is the smallest number you must add to 354 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
9. 9.Which of these numbers is divisible by 9? (a) 1491 (b) 1478 (c) 1496 (d) 1485
10. 10.What is the units digit of 7^27? (a) 1 (b) 9 (c) 7 (d) 3
11. 11.What is the units digit of 8^53? (a) 2 (b) 8 (c) 4 (d) 6
12. 12.What is the smallest number you must add to 367 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
13. 13.What is the remainder when 275 is divided by 12? (a) 11 (b) 9 (c) 10 (d) 1
14. 14.What is the remainder when 631 is divided by 8? (a) 5 (b) 1 (c) 6 (d) 7
15. 15.Which of these numbers is divisible by 8? (a) 3576 (b) 3575 (c) 3586 (d) 3582
16. 16.What is the units digit of 8^43? (a) 4 (b) 8 (c) 2 (d) 6
17. 17.What is the smallest number you must add to 326 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 5
18. 18.What is the smallest number you must add to 128 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 2
19. 19.Which of these numbers is divisible by 6? (a) 767 (b) 756 (c) 757 (d) 755
20. 20.Which of these numbers is divisible by 11? (a) 908 (b) 922 (c) 913 (d) 927
21. 21.What is the units digit of 4^55? (a) 3 (b) 6 (c) 4 (d) 5
22. 22.What is the units digit of 9^56? (a) 0 (b) 9 (c) 1 (d) 2
23. 23.What is the remainder when 203 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
24. 24.Which of these numbers is divisible by 4? (a) 3557 (b) 3551 (c) 3552 (d) 3553
25. 25.Which of these numbers is divisible by 3? (a) 622 (b) 621 (c) 625 (d) 619
26. 26.What is the units digit of 9^39? (a) 1 (b) 9 (c) 8 (d) 7
27. 27.What is the units digit of 3^18? (a) 7 (b) 1 (c) 3 (d) 9
28. 28.Which of these numbers is divisible by 4? (a) 253 (b) 261 (c) 256 (d) 258
29. 29.Which of these numbers is divisible by 11? (a) 5519 (b) 5499 (c) 5500 (d) 5501
30. 30.What is the smallest number you must add to 74 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 4
31. 31.What is the smallest number you must add to 124 to make it divisible by 3? (a) 1 (b) 2
32. 32.What is the smallest number you must add to 219 to make it divisible by 12? (a) 8 (b) 9 (c) 10 (d) 3
33. 33.What is the smallest number you must add to 274 to make it divisible by 8? (a) 6 (b) 2 (c) 5 (d) 7
34. 34.Which of these numbers is divisible by 3? (a) 2475 (b) 2480 (c) 2473 (d) 2477
35. 35.What is the remainder when 195 is divided by 12? (a) 4 (b) 3 (c) 2 (d) 9
36. 36.What is the remainder when 324 is divided by 8? (a) 5 (b) 3 (c) 6 (d) 4
37. 37.What is the remainder when 263 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
38. 38.What is the smallest number you must add to 281 to make it divisible by 7? (a) 5 (b) 1 (c) 6 (d) 4
39. 39.What is the units digit of 3^13? (a) 3 (b) 9 (c) 7 (d) 1
40. 40.Which of these numbers is divisible by 8? (a) 1951 (b) 1962 (c) 1955 (d) 1952

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