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Home » #### \(2\sqrt29\) - Professionaloutdoormedia

#### \(2\sqrt29\) - Professionaloutdoormedia

📅 May 13, 2026 👤 scraface

May 13, 2026

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📌 Solution: We are given the system of congruences:

📌 We solve this using the method of successive substitutions or the Chinese Remainder Theorem. From the first congruence, write:

📌 We now find the modular inverse of 7 modulo 9. Testing small values:

📌 Substitute back into $ n = 7k + 3 $:

📌 n = 7(9m + 4) + 3 = 63m + 28 + 3 = 63m + 31

📌 The smallest positive $ n $ occurs when $ m = 0 $:

📌 Question: A palynologist is categorizing pollen samples and finds that a particular type appears in clusters that are multiples of 18 and 24, but not in clusters that are multiples of 30. What is the smallest such cluster size greater than 100?

📌 Solution: We are looking for the smallest integer greater than 100 that is a common multiple of 18 and 24, but not a multiple of 30. First, compute the least common multiple (LCM) of 18 and 24:

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Equality occurs when $ \frac{x}{y} = \frac{y}{z} = \frac{z}{x} $, whic...
\frac{(1/3)^2}{1/3} + \frac{(1/3)^2}{1/3} + \frac{(1/3)^2}{1/3} = 3 \c...
Therefore, the minimum value is $ \boxed{1} $.
Question: Compute the sum of the roots of the equation
u\sqrt{u} - 6u + 7\sqrt{u} - 12 = 0,
given that all roots are real and non-negative.
Solution: Let $ \sqrt{u} = t $, so $ u = t^2 $. Substituting into the...
t^2 \cdot t - 6t^2 + 7t - 12 = 0 \Rightarrow t^3 - 6t^2 + 7t - 12 = 0
We are now to find the sum of the roots of the cubic equation
t^3 - 6t^2 + 7t - 12 = 0

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