#### 9.7Question: What is the largest possible value of $\gcd(a,b)$ if the sum of two positive integers $a$ and $b$ is 2025, representing the total number of seismic tremors recorded over two consecutive years?

["# What Is the Largest Possible Value of $\gcd(a,b)$ When $a + b = 2025$?", "When tracking seismic activity, researchers often record total tremor counts over consecutive years. Suppose the combined number of seismic events over two years is exactly 2025 — a number that invites mathematical exploration. A particularly interesting question arises: What is the largest possible value of $\gcd(a,b)$, where $a$ and $b$ are the number of tremors recorded in each year and $a + b = 2025$?", "In number theory, the greatest common divisor (gcd) of two positive integers $a$ and $b$ that sum to a fixed total reveals deep structure. Let $d = \gcd(a,b)$. Then we can write $a = d \cdot m$ and $b = d \cdot n$, where $m$ and $n$ are coprime integers ($\gcd(m,n) = 1$). Since $a + b = 2025$, we have:", "$$\nd(m + n) = 2025\n$$", "This implies that $d$ must be a divisor of 2025, and $m+n = \frac{2025}{d}$. Our goal is to maximize $d$, subject to the condition that $m$ and $n$ are positive coprime integers summing to $\frac{2025}{d}$.", "### Step 1: Factorize 2025", "Begin by factoring 2025:\n$$\n2025 = 25 \ imes 81 = 5^2 \ imes 3^4\n$$", "So, 2025 = $3^4 \cdot 5^2$. The total number of positive divisors is $(4+1)(2+1) = 15$, giving manageable candidates for $d$.", "### Step 2: Maximize $d$ such that $\frac{2025}{d} = m+n$ has coprime positive integers $m$ and $n$", "For any integer $s = m+n$, can we always find coprime positive integers $m$ and $n$ with $m+n = s$? Yes — for any $s \geq 2$, the pair $m=1$, $n=s-1$ satisfies $\gcd(1, s-1) = 1$. Thus, as long as $s = \frac{2025}{d} \geq 2$, we can always choose $m=1$, $n=s-1$, and $\gcd(m,n)=1$.", "Therefore, the only requirement to maximize $d$ is that $d$ divides 2025 and $\frac{2025}{d} \geq 2$. Since the smallest valid $m+n$ is 2, the largest possible $d$ occurs when $s = \frac{2025}{d} = 2$, provided 2 divides 2025.", "But 2025 is odd, so 2 does not divide 2025. The next smallest possible $s = m+n$ that divides 2025 and is at least 2 must be the smallest odd divisor of 2025 greater than 1.", "List small divisors of 2025:\n1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, 2025", "We want the largest $d$ such that $\frac{2025}{d} \geq 2$ and $d \mid 2025$. Since $m+n = \frac{2025}{d} \geq 2$, we minimize $s = m+n$ to maximize $d$.", "Try smallest valid $s \geq 2$: $s = 3$", "Then $d = \frac{2025}{3} = 675$", "Check: Is $s=3$ sufficient? Yes — pick $m=1$, $n=2$, $\gcd(1,2)=1$, so valid.", "Can we go higher? The next smallest divisor greater than 3 is 5 → $d = 2025 / 5 = 405 < 675$", "Larger divisors give smaller $d$.", "Thus, the maximum occurs at $s = 3$, $d = 675$", "### Step 3: Verify the solution", "Let $d = 675$, then $a = 675 \cdot 1 = 675$, $b = 675 \cdot 2 = 1350$, and $a + b = 2025$", "$\gcd(675, 1350) = 675$, and $\gcd(1,2)=1$, so valid.", "Any larger $d$ would require $\frac{2025}{d} < 3$, but the next divisor down is 405 → $m+n=5$, giving $d=405 < 675$", "Hence, 675 is indeed the maximum.", "### Conclusion", "When $a + b = 2025$, the largest possible value of $\gcd(a,b)$ is:", "$$\n\boxed{675}\n$$", "This result has practical relevance in analyzing seismic data — identifying common underlying factors in tremor patterns across years helps detect systemic geophysical trends. Understanding maximum gcd values enables modeling of correlated event clusters.", "For two years with a total of 2025 seismic events, the highest possible shared gcd of event counts per year is $\boxed{675}$, achieved when events are in a 1:2 ratio."]