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Home » \gcd(5^m - 1, 5^n - 1) = 5^\gcd(m,n) - 1. - Ready Digital AB

\gcd(5^m - 1, 5^n - 1) = 5^\gcd(m,n) - 1. - Ready Digital AB

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 So \(1 \leq y \leq rac{5}{3}\)

📌 Check if all values are attainable: at \(\sin x = 1\), \(f = rac{2+3}{1+2} = rac{5}{3}\), at \(\sin x = -1\), \(f = rac{-2+3}{-1+2} = rac{1}{1} = 1\). Intermediate values follow continuously.

📌 oxed{\left[1,\ rac{5}{3}

📌 Question: Find the matrix \(\mathbf{M}\) such that \(\mathbf{M} egin{bmatrix} 1 \ 2 \end{bmatrix} = egin{bmatrix} 4 \ 1 \end{bmatrix}\) and \(\mathbf{M} egin{bmatrix} 3 \ -1 \end{bmatrix} = egin{bmatrix} 5 \ -2 \end{bmatrix}\).

📌 Solution: Let \(\mathbf{M} = egin{bmatrix} a & b \ c & d \end{bmatrix}\)

📌 First equation:

📌 egin{bmatrix} a & b \ c & d \end{bmatrix} egin{bmatrix} 1 \ 2 \end{bmatrix} = egin{bmatrix} a + 2b \ c + 2d \end{bmatrix} = egin{bmatrix} 4 \ 1 \end{bmatrix}

📌 \Rightarrow

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