Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 409 · 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 6? (a) 1256 (b) 1249 (c) 1254 (d) 1262
2. 2.What is the remainder when 469 is divided by 6? (a) 2 (b) 1 (c) 5 (d) 0
3. 3.What is the smallest number you must add to 157 to make it divisible by 10? (a) 3 (b) 4 (c) 7 (d) 2
4. 4.What is the smallest number you must add to 534 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
5. 5.Which of these numbers is divisible by 4? (a) 3467 (b) 3464 (c) 3471 (d) 3463
6. 6.What is the units digit of 9^38? (a) 0 (b) 2 (c) 1 (d) 9
7. 7.Which of these numbers is divisible by 8? (a) 6110 (b) 6100 (c) 6104 (d) 6115
8. 8.Which of these numbers is divisible by 4? (a) 697 (b) 706 (c) 701 (d) 700
9. 9.What is the units digit of 8^15? (a) 2 (b) 4 (c) 6 (d) 8
10. 10.What is the smallest number you must add to 207 to make it divisible by 7? (a) 2 (b) 3 (c) 4 (d) 5
11. 11.What is the units digit of 8^19? (a) 6 (b) 2 (c) 8 (d) 4
12. 12.What is the remainder when 68 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
13. 13.What is the smallest number you must add to 41 to make it divisible by 4? (a) 2 (b) 1 (c) 3
14. 14.What is the units digit of 8^23? (a) 8 (b) 6 (c) 2 (d) 4
15. 15.What is the units digit of 3^31? (a) 3 (b) 9 (c) 1 (d) 7
16. 16.What is the units digit of 3^39? (a) 9 (b) 3 (c) 1 (d) 7
17. 17.What is the units digit of 2^51? (a) 4 (b) 2 (c) 8 (d) 6
18. 18.What is the units digit of 7^54? (a) 3 (b) 7 (c) 1 (d) 9
19. 19.What is the units digit of 4^54? (a) 5 (b) 7 (c) 6 (d) 4
20. 20.What is the remainder when 303 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
21. 21.What is the remainder when 232 is divided by 7? (a) 2 (b) 0 (c) 1 (d) 6
22. 22.What is the remainder when 243 is divided by 7? (a) 4 (b) 5 (c) 2 (d) 6
23. 23.Which of these numbers is divisible by 8? (a) 5816 (b) 5810 (c) 5823 (d) 5829
24. 24.What is the remainder when 347 is divided by 7? (a) 3 (b) 4 (c) 6 (d) 5
25. 25.Which of these numbers is divisible by 11? (a) 500 (b) 506 (c) 513 (d) 525
26. 26.What is the smallest number you must add to 195 to make it divisible by 9? (a) 3 (b) 4 (c) 6 (d) 2
27. 27.What is the smallest number you must add to 332 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
28. 28.What is the smallest number you must add to 651 to make it divisible by 10? (a) 1 (b) 7 (c) 8 (d) 9
29. 29.Which of these numbers is divisible by 3? (a) 1573 (b) 1568 (c) 1569 (d) 1570
30. 30.What is the smallest number you must add to 80 to make it divisible by 3? (a) 2 (b) 1
31. 31.What is the units digit of 9^10? (a) 0 (b) 9 (c) 2 (d) 1
32. 32.What is the remainder when 214 is divided by 8? (a) 6 (b) 5 (c) 2 (d) 7
33. 33.What is the units digit of 7^32? (a) 3 (b) 7 (c) 9 (d) 1
34. 34.What is the remainder when 363 is divided by 12? (a) 2 (b) 3 (c) 9 (d) 4
35. 35.What is the smallest number you must add to 382 to make it divisible by 6? (a) 2 (b) 4 (c) 3 (d) 1
36. 36.What is the units digit of 9^14? (a) 9 (b) 1 (c) 0 (d) 2
37. 37.What is the smallest number you must add to 513 to make it divisible by 7? (a) 6 (b) 4 (c) 2 (d) 5
38. 38.What is the remainder when 53 is divided by 3? (a) 1 (b) 2 (c) 0
39. 39.Which of these numbers is divisible by 3? (a) 1502 (b) 1497 (c) 1495 (d) 1499
40. 40.Which of these numbers is divisible by 9? (a) 7424 (b) 7431 (c) 7416 (d) 7415

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