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Home » الحل: نطبق خوارزمية إقليدس لإيجاد $\gcd(105, 165)$. - AssociationVoting

الحل: نطبق خوارزمية إقليدس لإيجاد $\gcd(105, 165)$. - AssociationVoting

📅 May 15, 2026 👤 scraface

May 15, 2026

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📌 La condición $ X \leq Y - 15 $ implica $ Y \geq X + 15 $. La región en el cuadrado que satisface esto está delimitada por la línea $ Y = X + 15 $, intersectada con $ Y \leq 60 $.

📌 Enjulga los límites:

📌 Cuando $ X = 0 $, $ Y \geq 15 $

📌 Cuando $ X = 45 $, $ Y \geq 60 $

📌 Para $ X > 45 $, $ Y \geq X + 15 > 60 $, lo cual está fuera del cuadrado

📌 Así, la región válida es un trapecio desde $ X = 0 $ hasta $ X = 45 $, con altura constante en $ Y $ desde $ X+15 $ hasta 60.

📌 Área de esta región:

📌 \int_0^{45} (60 - (X + 15))\, dX = \int_0^{45} (45 - X)\, dX = \left[45X - \frac{X^2}{2}\right]_0^{45} = 45 \cdot 45 - \frac{45^2}{2} = 2025 - \frac{2025}{2} = \frac{2025}{2}

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