Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 770 · 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 remainder when 59 is divided by 3? (a) 0 (b) 2 (c) 1
2. 2.Which of these numbers is divisible by 11? (a) 7754 (b) 7768 (c) 7755 (d) 7757
3. 3.What is the remainder when 313 is divided by 7? (a) 4 (b) 6 (c) 2 (d) 5
4. 4.What is the smallest number you must add to 647 to make it divisible by 11? (a) 3 (b) 2 (c) 9 (d) 1
5. 5.What is the smallest number you must add to 104 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 4
6. 6.What is the smallest number you must add to 760 to make it divisible by 11? (a) 1 (b) 10 (c) 9 (d) 8
7. 7.What is the smallest number you must add to 349 to make it divisible by 7? (a) 3 (b) 2 (c) 6 (d) 1
8. 8.What is the units digit of 3^47? (a) 3 (b) 1 (c) 7 (d) 9
9. 9.What is the remainder when 199 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
10. 10.What is the remainder when 546 is divided by 9? (a) 3 (b) 6 (c) 7 (d) 5
11. 11.What is the smallest number you must add to 248 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
12. 12.What is the units digit of 9^46? (a) 1 (b) 2 (c) 0 (d) 9
13. 13.What is the smallest number you must add to 287 to make it divisible by 10? (a) 7 (b) 4 (c) 3 (d) 2
14. 14.Which of these numbers is divisible by 6? (a) 1407 (b) 1421 (c) 1412 (d) 1410
15. 15.What is the remainder when 444 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 3
16. 16.Which of these numbers is divisible by 11? (a) 5241 (b) 5257 (c) 5247 (d) 5259
17. 17.What is the smallest number you must add to 187 to make it divisible by 7? (a) 2 (b) 1 (c) 5 (d) 3
18. 18.What is the units digit of 7^50? (a) 1 (b) 9 (c) 3 (d) 7
19. 19.What is the remainder when 249 is divided by 11? (a) 7 (b) 8 (c) 6 (d) 4
20. 20.What is the units digit of 3^19? (a) 9 (b) 3 (c) 1 (d) 7
21. 21.What is the units digit of 7^19? (a) 1 (b) 9 (c) 3 (d) 7
22. 22.What is the units digit of 8^48? (a) 8 (b) 6 (c) 4 (d) 2
23. 23.What is the units digit of 8^28? (a) 8 (b) 6 (c) 2 (d) 4
24. 24.Which of these numbers is divisible by 6? (a) 435 (b) 429 (c) 424 (d) 426
25. 25.What is the smallest number you must add to 81 to make it divisible by 4? (a) 2 (b) 1 (c) 3
26. 26.What is the smallest number you must add to 614 to make it divisible by 9? (a) 2 (b) 6 (c) 8 (d) 7
27. 27.Which of these numbers is divisible by 11? (a) 6967 (b) 6988 (c) 6982 (d) 6974
28. 28.What is the remainder when 570 is divided by 11? (a) 2 (b) 8 (c) 10 (d) 9
29. 29.Which of these numbers is divisible by 6? (a) 3602 (b) 3613 (c) 3606 (d) 3609
30. 30.What is the units digit of 2^18? (a) 6 (b) 2 (c) 8 (d) 4
31. 31.Which of these numbers is divisible by 8? (a) 1697 (b) 1717 (c) 1704 (d) 1709
32. 32.What is the units digit of 8^24? (a) 4 (b) 2 (c) 8 (d) 6
33. 33.Which of these numbers is divisible by 6? (a) 5259 (b) 5254 (c) 5267 (d) 5256
34. 34.What is the units digit of 8^16? (a) 8 (b) 4 (c) 2 (d) 6
35. 35.Which of these numbers is divisible by 3? (a) 2065 (b) 2059 (c) 2063 (d) 2061
36. 36.What is the remainder when 397 is divided by 11? (a) 2 (b) 1 (c) 10 (d) 0
37. 37.Which of these numbers is divisible by 8? (a) 6589 (b) 6582 (c) 6576 (d) 6575
38. 38.Which of these numbers is divisible by 9? (a) 1778 (b) 1799 (c) 1782 (d) 1788
39. 39.What is the remainder when 206 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
40. 40.What is the smallest number you must add to 93 to make it divisible by 4? (a) 1 (b) 3 (c) 2

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