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Консультация № 200178
28.01.2021, 18:14
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Здравствуйте! Прошу помощи в следующем вопросе:
Задание 4. Натуральные числа m и n таковы, что дробь m/n
несократимая, а дробь 4m+3n/5m+2n сократима. На какие натуральные числа она сокращается?
Задание 4. Натуральные числа m и n таковы, что дробь m/n
несократимая, а дробь 4m+3n/5m+2n сократима. На какие натуральные числа она сокращается?
Обсуждение
01.02.2021, 12:53
общий
это ответ
Здравствуйте, kuznetsova.79!
Положим n = m + k. Так как m/n несократима, m и k не имеют общих натуральных делителей. Подставим:
(4m + 3n)/(5m + 2n) = (4m + 3m + 3k)/(5m + 2m + 2k) = (7m + 3k)/(7m + 2k) = 1 + k/(7m + 2k).
Исходная дробь сократима тогда и только тогда, когда сократима дробь k/(7m + k). Так как m и k не имеют общих делителей, дробь можно сократить только на 7.
Положим n = m + k. Так как m/n несократима, m и k не имеют общих натуральных делителей. Подставим:
(4m + 3n)/(5m + 2n) = (4m + 3m + 3k)/(5m + 2m + 2k) = (7m + 3k)/(7m + 2k) = 1 + k/(7m + 2k).
Исходная дробь сократима тогда и только тогда, когда сократима дробь k/(7m + k). Так как m и k не имеют общих делителей, дробь можно сократить только на 7.
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