Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 308 · 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 4? (a) 1601 (b) 1597 (c) 1596 (d) 1595
2. 2.What is the remainder when 755 is divided by 12? (a) 9 (b) 11 (c) 1 (d) 10
3. 3.What is the smallest number you must add to 424 to make it divisible by 7? (a) 5 (b) 2 (c) 3 (d) 4
4. 4.What is the remainder when 206 is divided by 7? (a) 2 (b) 5 (c) 4 (d) 3
5. 5.What is the smallest number you must add to 210 to make it divisible by 4? (a) 1 (b) 3 (c) 2
6. 6.What is the remainder when 338 is divided by 8? (a) 6 (b) 3 (c) 2 (d) 1
7. 7.What is the remainder when 606 is divided by 11? (a) 10 (b) 1 (c) 2 (d) 0
8. 8.What is the smallest number you must add to 199 to make it divisible by 3? (a) 1 (b) 2
9. 9.What is the smallest number you must add to 104 to make it divisible by 3? (a) 1 (b) 2
10. 10.What is the smallest number you must add to 149 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
11. 11.Which of these numbers is divisible by 6? (a) 5223 (b) 5228 (c) 5236 (d) 5226
12. 12.What is the smallest number you must add to 317 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
13. 13.Which of these numbers is divisible by 8? (a) 6087 (b) 6075 (c) 6089 (d) 6080
14. 14.What is the remainder when 267 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
15. 15.What is the remainder when 604 is divided by 9? (a) 1 (b) 2 (c) 0 (d) 8
16. 16.Which of these numbers is divisible by 8? (a) 6922 (b) 6920 (c) 6916 (d) 6935
17. 17.What is the remainder when 259 is divided by 10? (a) 9 (b) 1 (c) 7 (d) 8
18. 18.Which of these numbers is divisible by 9? (a) 7407 (b) 7402 (c) 7423 (d) 7413
19. 19.What is the units digit of 8^49? (a) 8 (b) 6 (c) 4 (d) 2
20. 20.What is the remainder when 164 is divided by 6? (a) 4 (b) 1 (c) 2 (d) 3
21. 21.What is the units digit of 9^22? (a) 0 (b) 2 (c) 1 (d) 9
22. 22.What is the smallest number you must add to 257 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
23. 23.What is the smallest number you must add to 67 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
24. 24.Which of these numbers is divisible by 6? (a) 2500 (b) 2507 (c) 2510 (d) 2502
25. 25.What is the units digit of 7^38? (a) 1 (b) 7 (c) 9 (d) 3
26. 26.What is the remainder when 86 is divided by 3? (a) 1 (b) 2 (c) 0
27. 27.What is the remainder when 577 is divided by 10? (a) 3 (b) 6 (c) 8 (d) 7
28. 28.What is the remainder when 143 is divided by 7? (a) 3 (b) 5 (c) 4 (d) 2
29. 29.What is the remainder when 46 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
30. 30.What is the units digit of 7^59? (a) 3 (b) 9 (c) 1 (d) 7
31. 31.What is the remainder when 351 is divided by 7? (a) 6 (b) 1 (c) 0 (d) 2
32. 32.What is the smallest number you must add to 185 to make it divisible by 10? (a) 5 (b) 4 (c) 6 (d) 7
33. 33.Which of these numbers is divisible by 3? (a) 781 (b) 780 (c) 785 (d) 778
34. 34.What is the remainder when 38 is divided by 3? (a) 0 (b) 1 (c) 2
35. 35.What is the smallest number you must add to 762 to make it divisible by 11? (a) 7 (b) 9 (c) 8 (d) 3
36. 36.Which of these numbers is divisible by 9? (a) 3561 (b) 3564 (c) 3581 (d) 3569
37. 37.Which of these numbers is divisible by 8? (a) 322 (b) 313 (c) 320 (d) 330
38. 38.Which of these numbers is divisible by 3? (a) 1921 (b) 1924 (c) 1923 (d) 1928
39. 39.What is the smallest number you must add to 145 to make it divisible by 3? (a) 1 (b) 2
40. 40.Which of these numbers is divisible by 9? (a) 2575 (b) 2585 (c) 2583 (d) 2594

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