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📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 We are given the ratio of red to clay particles is \( 5:3 \), and there are 15 red particles. Let the number of clay particles be \( c \). Set up the proportion:

📌 \frac{5}{3} = \frac{15}{c}

📌 5c = 45 \Rightarrow c = \frac{45}{5} = 9

📌 Thus, the number of clay particles is \(\boxed{9}\).Question: A hydrologist is analyzing rainfall data over 7 consecutive days. If the total rainfall recorded is 140 mm, and each day's rainfall is a positive integer in millimeters, what is the maximum possible value of the least common multiple of the 7 daily rainfalls?

📌 To maximize the least common multiple (LCM) of 7 positive integers that sum to 140, we aim to distribute the total among the 7 numbers in such a way that their LCM is maximized. The LCM of a set of numbers is maximized when the numbers are pairwise coprime or have high prime powers.

📌 We begin by testing a configuration where the numbers are distinct primes or mostly coprime small integers. However, since the sum is fixed at 140 and we have 7 numbers, we cannot use 7 large primes (their sum would likely exceed 140). We consider using small numbers that are pairwise coprime and whose LCM is large.

📌 Try the sequence:

📌 1, 2, 3, 4, 5, 6, 120

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