the frog - Dygne

So available: 3, 4, 5, 6, 7, excluding 2 and 8 → valid: 3–7.

Apr 21, 2026

We need 2 non-adjacent from {3,4,5,6,7}.

Apr 21, 2026

List all 2-element non-adjacent pairs in 5 consecutive positions:

Apr 21, 2026

Total: 6 valid pairs with station 1 selected.

Apr 21, 2026

But each selection is counted 3 times (once for each selected station), so total valid sets:

Apr 21, 2026

\frac{8 \times 6}{3} = 16

Apr 21, 2026

Thus, number of non-adjacent triplets is 16.

Apr 21, 2026

\frac{16}{56} = \frac{2}{7}

Apr 21, 2026

The probability that no two selected stations are adjacent is $\boxed{\frac{2}{7}}$.Question: What is the largest possible value of $\gcd(a,b)$ if the sum of two positive integers $a$ and $b$ is 2025, representing the years of two tectonic shifts?

Apr 21, 2026

Sكية: The greatest common divisor of $a$ and $b$ must divide their sum, which is 2025. To maximize $\gcd(a,b)$, we find the largest proper divisor of 2025. Factoring 2025: $2025 = 3^4 \times 5^2$. The largest proper divisor is $2025 / 3 = 675$. Thus, the maximum $\gcd(a,b)$ is $\boxed{675}$.

Apr 21, 2026