2.63 Найдите наибольший общий делитель чисел: а) 45, 60 и 105; б) 162, 222 и 432; в) 108, 72 и 96; г) 240, 480 и 720.

Ответ:

Чтобы найти наибольший общий делитель (НОД) нескольких чисел, нужно разложить каждое из них на простые множители и найти произведение общих множителей.

а) 45, 60 и 105:

45 = 3 \(\times\) 3 \(\times\) 5

60 = 2 \(\times\) 2 \(\times\) 3 \(\times\) 5

105 = 3 \(\times\) 5 \(\times\) 7

НОД(45, 60, 105) = 3 \(\times\) 5 = 15

б) 162, 222 и 432:

162 = 2 \(\times\) 3 \(\times\) 3 \(\times\) 3 \(\times\) 3

222 = 2 \(\times\) 3 \(\times\) 37

432 = 2 \(\times\) 2 \(\times\) 2 \(\times\) 2 \(\times\) 3 \(\times\) 3 \(\times\) 3

НОД(162, 222, 432) = 2 \(\times\) 3 = 6

в) 108, 72 и 96:

108 = 2 \(\times\) 2 \(\times\) 3 \(\times\) 3 \(\times\) 3

72 = 2 \(\times\) 2 \(\times\) 2 \(\times\) 3 \(\times\) 3

96 = 2 \(\times\) 2 \(\times\) 2 \(\times\) 2 \(\times\) 2 \(\times\) 3

НОД(108, 72, 96) = 2 \(\times\) 2 \(\times\) 3 = 12

г) 240, 480 и 720:

240 = 2 \(\times\) 2 \(\times\) 2 \(\times\) 2 \(\times\) 3 \(\times\) 5

480 = 2 \(\times\) 2 \(\times\) 2 \(\times\) 2 \(\times\) 2 \(\times\) 3 \(\times\) 5

720 = 2 \(\times\) 2 \(\times\) 2 \(\times\) 2 \(\times\) 3 \(\times\) 3 \(\times\) 5

НОД(240, 480, 720) = 2 \(\times\) 2 \(\times\) 2 \(\times\) 2 \(\times\) 3 \(\times\) 5 = 240