A philosopher of science is contemplating the symmetry of natural laws and wonders: If the sum of two positive integers \(a\) and \(b\) equals 2023, what is the largest possible value of \(\gcd(a, b)\)?
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- Area of one equilateral triangle: \(\frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3}\).
- Total area = \(6 \times 9\sqrt{3} = 54\sqrt{3}\) cm².
- #### \(54\sqrt{3}\) cm²**Question:
- To maximize \(\gcd(a, b)\), we note that if \(\gcd(a, b) = d\), then \(a = dx\) and \(b = dy\) for some integers \(x\) and \(y\) with \(\gcd(x, y) = 1\). Thus, \(a + b = dx + dy = d(x + y) = 2023\). Therefore, \(d\) must divide 2023.
- First, we factor 2023 to find its divisors. Checking small prime numbers:
- 2023 is odd, so not divisible by 2.
