Question: What is the largest possible value of $\gcd(a, b)$ if $a + b = 100$ and $a, b$ are positive integers?
["Understanding the Largest Possible Value of $\gcd(a, b)$ When $a + b = 100$", "When two positive integers $a$ and $b$ sum to 100, their greatest common divisor (gcd), denoted $\gcd(a, b)$, plays an important role in number theory and has practical relevance in problems involving divisors and integer partitions. For anyone asking, “What is the largest possible value of $\gcd(a, b)$ if $a + b = 100$?”, the answer lies not just in a numeric value, but in the deeper mathematical insight behind it.", "### What Is $\gcd(a, b)$ and Why Does It Matter?", "The greatest common divisor of two integers $a$ and $b$ is the largest positive integer that divides both $a$ and $b$ without a remainder. If $d = \gcd(a, b)$, then $a = d \cdot m$ and $b = d \cdot n$ for some positive integers $m$ and $n$ with $\gcd(m, n) = 1$. Substituting into the sum gives:", "[\na + b = d \cdot m + d \cdot n = d(m + n) = 100\n]", "Thus, $d$ must be a divisor of 100. The largest possible value of $d$ occurs when $m + n$ is minimized. Since $m$ and $n$ are coprime positive integers, the smallest sum $m + n$ can be is 2 — achieved when $m = n = 1$. This corresponds to $a = b = d$, so:", "[\nd + d = 100 \Rightarrow 2d = 100 \Rightarrow d = 50\n]", "### Can $\gcd(a, b) = 50 Be Achieved?", "Yes. Let $a = 50$ and $b = 50$. Both are positive integers, $a + b = 100$, and $\gcd(50, 50) = 50$. Since $m = n = 1$, which are coprime, this satisfies all conditions. Thus, $d = 50$ is attainable.", "### Is a Larger $\gcd$ Possible?", "Suppose $\gcd(a, b) = d > 50$. Then $d$ must divide 100, and the only divisors greater than 50 are 100 itself. But if $\gcd(a, b) = 100$, then both $a$ and $b$ would be multiples of 100. Since $a + b = 100$, the only possibility would be $a = 100$, $b = 0$ — but $b$ must be a positive integer, so this is invalid. Therefore, $\gcd(a, b)$ cannot exceed 50 when both $a$ and $b$ are positive integers summing to 100.", "### Summary: The Maximum $\gcd(a, b)$ Is 50", "The largest possible value of $\gcd(a, b)$ given $a + b = 100$ with $a, b$ positive integers is:", "[\n\boxed{50}\n]", "This maximum occurs when $a = b = 50$, and reflects the interplay between divisors, coprimality, and integer sums — fundamental concepts in number theory and discrete mathematics.", "For students, puzzle solvers, or anyone exploring divisibility, recognizing that pairing $a$ and $b$ as multiples of their gcd leads directly to the solution, simplifying otherwise complex investigations.", "---", "TL;DR: The largest possible value of $\gcd(a, b)$ when $a + b = 100$ and $a, b$ are positive integers is 50, achieved when $a = b = 50$. This occurs because $100 = 2 \ imes 50$, and both numbers are equal multiples of 50, which is coprime to itself, maximizing the gcd."]
