Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 666 · 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 units digit of 3^53? (a) 1 (b) 3 (c) 7 (d) 9
2. 2.What is the remainder when 230 is divided by 12? (a) 1 (b) 3 (c) 10 (d) 2
3. 3.Which of these numbers is divisible by 4? (a) 3427 (b) 3421 (c) 3419 (d) 3420
4. 4.What is the remainder when 273 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
5. 5.Which of these numbers is divisible by 6? (a) 1063 (b) 1075 (c) 1072 (d) 1068
6. 6.What is the remainder when 326 is divided by 9? (a) 3 (b) 7 (c) 1 (d) 2
7. 7.What is the units digit of 2^52? (a) 2 (b) 6 (c) 4 (d) 8
8. 8.Which of these numbers is divisible by 4? (a) 467 (b) 469 (c) 462 (d) 464
9. 9.What is the units digit of 3^30? (a) 7 (b) 3 (c) 9 (d) 1
10. 10.What is the units digit of 3^47? (a) 1 (b) 3 (c) 9 (d) 7
11. 11.What is the remainder when 203 is divided by 3? (a) 0 (b) 1 (c) 2
12. 12.What is the units digit of 2^51? (a) 2 (b) 4 (c) 6 (d) 8
13. 13.What is the remainder when 577 is divided by 8? (a) 1 (b) 2 (c) 7 (d) 0
14. 14.What is the remainder when 329 is divided by 8? (a) 2 (b) 7 (c) 0 (d) 1
15. 15.Which of these numbers is divisible by 3? (a) 2591 (b) 2594 (c) 2589 (d) 2587
16. 16.What is the smallest number you must add to 51 to make it divisible by 4? (a) 3 (b) 1 (c) 2
17. 17.What is the remainder when 72 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
18. 18.What is the smallest number you must add to 496 to make it divisible by 9? (a) 8 (b) 1 (c) 6 (d) 7
19. 19.Which of these numbers is divisible by 8? (a) 560 (b) 565 (c) 571 (d) 554
20. 20.What is the remainder when 479 is divided by 8? (a) 5 (b) 6 (c) 1 (d) 7
21. 21.Which of these numbers is divisible by 8? (a) 879 (b) 886 (c) 890 (d) 880
22. 22.What is the smallest number you must add to 316 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 6
23. 23.What is the remainder when 75 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
24. 24.What is the units digit of 3^10? (a) 7 (b) 1 (c) 3 (d) 9
25. 25.Which of these numbers is divisible by 8? (a) 5759 (b) 5752 (c) 5762 (d) 5749
26. 26.Which of these numbers is divisible by 6? (a) 508 (b) 513 (c) 520 (d) 510
27. 27.What is the remainder when 633 is divided by 11? (a) 7 (b) 8 (c) 5 (d) 6
28. 28.What is the smallest number you must add to 314 to make it divisible by 10? (a) 6 (b) 4 (c) 7 (d) 5
29. 29.What is the units digit of 3^41? (a) 3 (b) 9 (c) 7 (d) 1
30. 30.Which of these numbers is divisible by 4? (a) 1887 (b) 1884 (c) 1891 (d) 1883
31. 31.What is the remainder when 311 is divided by 12? (a) 11 (b) 10 (c) 9 (d) 1
32. 32.What is the remainder when 43 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
33. 33.What is the units digit of 3^21? (a) 3 (b) 1 (c) 7 (d) 9
34. 34.What is the units digit of 7^41? (a) 9 (b) 1 (c) 3 (d) 7
35. 35.Which of these numbers is divisible by 8? (a) 6120 (b) 6135 (c) 6123 (d) 6118
36. 36.What is the units digit of 4^23? (a) 5 (b) 4 (c) 3 (d) 6
37. 37.What is the units digit of 8^43? (a) 8 (b) 4 (c) 2 (d) 6
38. 38.Which of these numbers is divisible by 8? (a) 7008 (b) 7011 (c) 7005 (d) 7018
39. 39.What is the remainder when 92 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
40. 40.What is the smallest number you must add to 467 to make it divisible by 8? (a) 5 (b) 4 (c) 6 (d) 3

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