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Консультация № 201994
27.12.2021, 15:27
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4
1
Здравствуйте! У меня возникли сложности с таким вопросом:
Найдите количество несократимых дробей среди $$\frac{1}{9624}, \frac{2}{9624}, \frac{3}{9624}, ..., \frac{9623}{9624}, \frac{9624}{9624}$$.
Найдите количество несократимых дробей среди $$\frac{1}{9624}, \frac{2}{9624}, \frac{3}{9624}, ..., \frac{9623}{9624}, \frac{9624}{9624}$$.
Обсуждение
27.12.2021, 16:32
общий
Адресаты:
Если я правильно разбираюсь в LaTeX - форматировании, то дроби имеют вид 1/9624, 2/9624, 3/9624, ..., 9624/9624. Тогда количество сократимых дробей равно количеству делителей 9624, за исключением одного: 1. 9624 = 2^3 * 3 * 401. Тогда количество делителей по известной формуле (3 + 1) * (1 + 1) * (1 + 1) = 16 и ответ к задаче равен 15.
27.12.2021, 19:06
общий
Адресаты:
Тут всё немного сложнее, поскольку несократимыми будут только те дроби, у которых числитель и знаменатель - взаимно простые числа, то есть имеют наибольший общий делитель (НОД), равный единице. Остальные дроби будут сократимыми, даже если числитель не является делителем знаменателя. Например, 9/9624 = 3/3208 или 16/9624 = 2/1203 - сократимы, хотя 9 и 16 - не делители 9624.
31.12.2021, 07:30
общий
это ответ
Найдём количество взаимно простых чисел с 9624, меньших данного. Именно столько будет несократимых дробей, так как если подставить взаимно простое число в числитель, то дробь будет несократимой. Теперь понятно, что данное количество будет функцией Эйлера от 9624:
[$966$](9624) = (23 - 22) [$183$] (31 - 30) [$183$] (4011 - 4010) = 3200.
Значит, количество несократимых дробей равно 3200
Ответ: 3200
[$966$](9624) = (23 - 22) [$183$] (31 - 30) [$183$] (4011 - 4010) = 3200.
Значит, количество несократимых дробей равно 3200
Ответ: 3200
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