Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 314 · 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 smallest number you must add to 376 to make it divisible by 7? (a) 3 (b) 2 (c) 5 (d) 1
2. 2.What is the smallest number you must add to 563 to make it divisible by 11? (a) 8 (b) 10 (c) 2 (d) 9
3. 3.Which of these numbers is divisible by 3? (a) 486 (b) 487 (c) 484 (d) 490
4. 4.What is the remainder when 622 is divided by 11? (a) 8 (b) 7 (c) 5 (d) 6
5. 5.What is the smallest number you must add to 171 to make it divisible by 4? (a) 1 (b) 3 (c) 2
6. 6.What is the smallest number you must add to 386 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
7. 7.What is the units digit of 7^49? (a) 3 (b) 7 (c) 9 (d) 1
8. 8.What is the units digit of 8^15? (a) 6 (b) 2 (c) 8 (d) 4
9. 9.What is the smallest number you must add to 394 to make it divisible by 7? (a) 2 (b) 5 (c) 6 (d) 4
10. 10.What is the units digit of 4^19? (a) 4 (b) 6 (c) 3 (d) 5
11. 11.Which of these numbers is divisible by 3? (a) 1721 (b) 1717 (c) 1723 (d) 1719
12. 12.What is the smallest number you must add to 218 to make it divisible by 10? (a) 3 (b) 8 (c) 1 (d) 2
13. 13.Which of these numbers is divisible by 4? (a) 3473 (b) 3468 (c) 3471 (d) 3467
14. 14.Which of these numbers is divisible by 11? (a) 7548 (b) 7534 (c) 7539 (d) 7535
15. 15.What is the remainder when 628 is divided by 9? (a) 8 (b) 6 (c) 2 (d) 7
16. 16.Which of these numbers is divisible by 11? (a) 5188 (b) 5170 (c) 5171 (d) 5169
17. 17.What is the units digit of 9^38? (a) 9 (b) 2 (c) 0 (d) 1
18. 18.What is the units digit of 3^19? (a) 7 (b) 9 (c) 3 (d) 1
19. 19.Which of these numbers is divisible by 8? (a) 5935 (b) 5937 (c) 5928 (d) 5925
20. 20.What is the units digit of 2^15? (a) 2 (b) 6 (c) 4 (d) 8
21. 21.What is the units digit of 3^21? (a) 1 (b) 3 (c) 9 (d) 7
22. 22.Which of these numbers is divisible by 9? (a) 3815 (b) 3807 (c) 3821 (d) 3803
23. 23.What is the remainder when 32 is divided by 3? (a) 2 (b) 1 (c) 0
24. 24.What is the units digit of 7^11? (a) 9 (b) 7 (c) 3 (d) 1
25. 25.What is the smallest number you must add to 263 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 2
26. 26.Which of these numbers is divisible by 3? (a) 1866 (b) 1871 (c) 1868 (d) 1864
27. 27.Which of these numbers is divisible by 6? (a) 2839 (b) 2846 (c) 2838 (d) 2833
28. 28.Which of these numbers is divisible by 8? (a) 912 (b) 909 (c) 926 (d) 917
29. 29.What is the units digit of 4^14? (a) 5 (b) 7 (c) 4 (d) 6
30. 30.Which of these numbers is divisible by 9? (a) 5838 (b) 5843 (c) 5828 (d) 5832
31. 31.What is the smallest number you must add to 199 to make it divisible by 11? (a) 9 (b) 8 (c) 10 (d) 1
32. 32.What is the units digit of 4^43? (a) 5 (b) 3 (c) 4 (d) 6
33. 33.What is the units digit of 2^54? (a) 8 (b) 2 (c) 6 (d) 4
34. 34.Which of these numbers is divisible by 4? (a) 1559 (b) 1555 (c) 1556 (d) 1562
35. 35.Which of these numbers is divisible by 9? (a) 672 (b) 675 (c) 686 (d) 682
36. 36.What is the smallest number you must add to 145 to make it divisible by 8? (a) 1 (b) 6 (c) 7 (d) 5
37. 37.What is the remainder when 147 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
38. 38.What is the units digit of 4^20? (a) 7 (b) 6 (c) 4 (d) 5
39. 39.What is the remainder when 697 is divided by 11? (a) 3 (b) 4 (c) 5 (d) 7
40. 40.What is the smallest number you must add to 532 to make it divisible by 9? (a) 1 (b) 7 (c) 6 (d) 8

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