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Home » We start with the equation for population growth: - ToelettAPP

We start with the equation for population growth: - ToelettAPP

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 نختبر مثالاً: $1 \cdot 2 \cdot 3 \cdot 4 = 24$، والذي ليس قابلاً للقسمة على $48$.

📌 إذن، أكبر عدد صحيح يجب أن يقسم حاصل ضرب أي أربعة أعداد صحيحة متتالية هو $\boxed{24}$.

📌 السؤال: ما هي أكبر قيمة ممكنة للقاسم المشترك الأكبر $(a,b)$، إذا كان مجموع عددين صحيحين موجبين $a + b = 96$؟

📌 الحل: ليكن $d = \gcd(a, b)$. إذن $a = dx$، $b = dy$، حيث $\gcd(x, y) = 1$.

📌 ثم $a + b = d(x + y) = 96$، لذا يجب أن يكون $d$ قاسمًا للعدد $96$.

📌 لتعظيم $d$، نختار أكبر قاسم للعدد $96$ بحيث يكون $x + y$ عددًا صحيحًا موجبًا، و$\gcd(x, y) = 1$.

📌 أكبر قاسم للعدد $96$ هو $48$، مع $x + y = 2$. الزوج $(x, y) = (1, 1)$ يحقق $\gcd(1,1)=1$.

📌 إذن، $a = 48 \cdot 1 = 48$، $b = 48 \cdot 1 = 48$، و$\gcd(a,b) = 48$.

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