56. Найдите наименьшее общее кратное чисел:

Решение:

Чтобы найти наименьшее общее кратное (НОК) двух чисел, нужно найти их разложение на простые множители и перемножить все простые множители, взяв их в наибольшей степени, в которой они встречаются в каждом из разложений.

1. 6 и 8: 6 = 2 \(\cdot\) 3; 8 = 2³. НОК(6, 8) = 2³ \(\cdot\) 3 = 8 \(\cdot\) 3 = 24.
2. 18 и 72: 18 = 2 \(\cdot\) 3²; 72 = 2³ \(\cdot\) 3². НОК(18, 72) = 2³ \(\cdot\) 3² = 8 \(\cdot\) 9 = 72.
3. 16 и 56: 16 = 2⁴; 56 = 2³ \(\cdot\) 7. НОК(16, 56) = 2⁴ \(\cdot\) 7 = 16 \(\cdot\) 7 = 112.
4. 10 и 15: 10 = 2 \(\cdot\) 5; 15 = 3 \(\cdot\) 5. НОК(10, 15) = 2 \(\cdot\) 3 \(\cdot\) 5 = 30.
5. 32 и 96: 32 = 2⁵; 96 = 2⁵ \(\cdot\) 3. НОК(32, 96) = 2⁵ \(\cdot\) 3 = 32 \(\cdot\) 3 = 96.
6. 30 и 50: 30 = 2 \(\cdot\) 3 \(\cdot\) 5; 50 = 2 \(\cdot\) 5². НОК(30, 50) = 2 \(\cdot\) 3 \(\cdot\) 5² = 2 \(\cdot\) 3 \(\cdot\) 25 = 150.
7. 25 и 50: 25 = 5²; 50 = 2 \(\cdot\) 5². НОК(25, 50) = 2 \(\cdot\) 5² = 2 \(\cdot\) 25 = 50.
8. 90 и 180: 90 = 2 \(\cdot\) 3² \(\cdot\) 5; 180 = 2² \(\cdot\) 3² \(\cdot\) 5. НОК(90, 180) = 2² \(\cdot\) 3² \(\cdot\) 5 = 4 \(\cdot\) 9 \(\cdot\) 5 = 180.
9. 75 и 150: 75 = 3 \(\cdot\) 5²; 150 = 2 \(\cdot\) 3 \(\cdot\) 5². НОК(75, 150) = 2 \(\cdot\) 3 \(\cdot\) 5² = 2 \(\cdot\) 3 \(\cdot\) 25 = 150.
10. 60 и 220: 60 = 2² \(\cdot\) 3 \(\cdot\) 5; 220 = 2² \(\cdot\) 5 \(\cdot\) 11. НОК(60, 220) = 2² \(\cdot\) 3 \(\cdot\) 5 \(\cdot\) 11 = 4 \(\cdot\) 3 \(\cdot\) 5 \(\cdot\) 11 = 660.
11. 13 и 7: 13 - простое число; 7 - простое число. НОК(13, 7) = 13 \(\cdot\) 7 = 91.
12. 54 и 81: 54 = 2 \(\cdot\) 3³; 81 = 3⁴. НОК(54, 81) = 2 \(\cdot\) 3⁴ = 2 \(\cdot\) 81 = 162.
13. 5 и 31: 5 - простое число; 31 - простое число. НОК(5, 31) = 5 \(\cdot\) 31 = 155.
14. 17 и 187: 17 - простое число; 187 = 11 \(\cdot\) 17. НОК(17, 187) = 11 \(\cdot\) 17 = 187.
15. 48 и 64: 48 = 2⁴ \(\cdot\) 3; 64 = 2⁶. НОК(48, 64) = 2⁶ \(\cdot\) 3 = 64 \(\cdot\) 3 = 192.
16. 30 и 90: 30 = 2 \(\cdot\) 3 \(\cdot\) 5; 90 = 2 \(\cdot\) 3² \(\cdot\) 5. НОК(30, 90) = 2 \(\cdot\) 3² \(\cdot\) 5 = 2 \(\cdot\) 9 \(\cdot\) 5 = 90.
17. 25 и 75: 25 = 5²; 75 = 3 \(\cdot\) 5². НОК(25, 75) = 3 \(\cdot\) 5² = 3 \(\cdot\) 25 = 75.
18. 180 и 270: 180 = 2² \(\cdot\) 3² \(\cdot\) 5; 270 = 2 \(\cdot\) 3³ \(\cdot\) 5. НОК(180, 270) = 2² \(\cdot\) 3³ \(\cdot\) 5 = 4 \(\cdot\) 27 \(\cdot\) 5 = 540.
19. 175 и 225: 175 = 5² \(\cdot\) 7; 225 = 3² \(\cdot\) 5². НОК(175, 225) = 3² \(\cdot\) 5² \(\cdot\) 7 = 9 \(\cdot\) 25 \(\cdot\) 7 = 1575.
20. 35 и 40: 35 = 5 \(\cdot\) 7; 40 = 2³ \(\cdot\) 5. НОК(35, 40) = 2³ \(\cdot\) 5 \(\cdot\) 7 = 8 \(\cdot\) 5 \(\cdot\) 7 = 280.
21. 12 и 30: 12 = 2² \(\cdot\) 3; 30 = 2 \(\cdot\) 3 \(\cdot\) 5. НОК(12, 30) = 2² \(\cdot\) 3 \(\cdot\) 5 = 4 \(\cdot\) 3 \(\cdot\) 5 = 60.
22. 21 и 49: 21 = 3 \(\cdot\) 7; 49 = 7². НОК(21, 49) = 3 \(\cdot\) 7² = 3 \(\cdot\) 49 = 147.
23. 55 и 88: 55 = 5 \(\cdot\) 11; 88 = 2³ \(\cdot\) 11. НОК(55, 88) = 2³ \(\cdot\) 5 \(\cdot\) 11 = 8 \(\cdot\) 5 \(\cdot\) 11 = 440.
24. 100 и 160: 100 = 2² \(\cdot\) 5²; 160 = 2⁵ \(\cdot\) 5. НОК(100, 160) = 2⁵ \(\cdot\) 5² = 32 \(\cdot\) 25 = 800.
25. 50 и 320: 50 = 2 \(\cdot\) 5²; 320 = 2⁶ \(\cdot\) 5. НОК(50, 320) = 2⁶ \(\cdot\) 5² = 64 \(\cdot\) 25 = 1600.
26. 90 и 120: 90 = 2 \(\cdot\) 3² \(\cdot\) 5; 120 = 2³ \(\cdot\) 3 \(\cdot\) 5. НОК(90, 120) = 2³ \(\cdot\) 3² \(\cdot\) 5 = 8 \(\cdot\) 9 \(\cdot\) 5 = 360.
27. 75 и 100: 75 = 3 \(\cdot\) 5²; 100 = 2² \(\cdot\) 5². НОК(75, 100) = 2² \(\cdot\) 3 \(\cdot\) 5² = 4 \(\cdot\) 3 \(\cdot\) 25 = 300.
28. 370 и 360: 370 = 2 \(\cdot\) 5 \(\cdot\) 37; 360 = 2³ \(\cdot\) 3² \(\cdot\) 5. НОК(370, 360) = 2³ \(\cdot\) 3² \(\cdot\) 5 \(\cdot\) 37 = 8 \(\cdot\) 9 \(\cdot\) 5 \(\cdot\) 37 = 13320.
29. 225 и 300: 225 = 3² \(\cdot\) 5²; 300 = 2² \(\cdot\) 3 \(\cdot\) 5². НОК(225, 300) = 2² \(\cdot\) 3² \(\cdot\) 5² = 4 \(\cdot\) 9 \(\cdot\) 25 = 900.
30. 30 и 27: 30 = 2 \(\cdot\) 3 \(\cdot\) 5; 27 = 3³. НОК(30, 27) = 2 \(\cdot\) 3³ \(\cdot\) 5 = 2 \(\cdot\) 27 \(\cdot\) 5 = 270.

Ответ: 24, 72, 112, 30, 96, 150, 50, 180, 150, 660, 91, 162, 155, 187, 192, 90, 75, 540, 1575, 280, 60, 147, 440, 800, 1600, 360, 300, 13320, 900, 270.