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Home » Therefore, the sum is \( \boxed{\sqrt51 - 1} \). - AIKO, infinite ways to autonomy.

Therefore, the sum is \( \boxed{\sqrt51 - 1} \). - AIKO, infinite ways to autonomy.

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 r = \frac{A}{s}

📌 where $A$ is the area of the triangle and $s$ is the semi-perimeter:

📌 s = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = 21

📌 Using Heron’s formula for area:

📌 r = \frac{84}{21} = 4

📌 $$Question: What is the largest possible value of $\gcd(a,b)$, if the sum of two positive integers $a$ and $b$ is $1000$ and both $a$ and $b$ are divisible by a perfect square greater than $1$?

📌 Let $d = \gcd(a,b)$. Then we can write $a = dx$, $b = dy$, where $\gcd(x,y) = 1$. Since $a + b = 1000$, we have:

📌 dx + dy = d(x + y) = 1000 \Rightarrow d \mid 1000

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