JEE Main 2025 — 4 April, Evening Shift — Question 42

$$m = 6a, n = 6b$$ $$\text{So } \gcd(m, n) = 6 \implies \gcd(a, b) = 1$$ $$m = 6a \geq 10 \implies a \geq \left\lceil \frac{10}{6} \right\rceil = 2$$ $$m = 6a < 99 \implies a < \left\lceil \frac{99}{6} \right\rceil = 16.5 \implies a \leq 16$$

$\text{So } a, b \in \{2, 3, \dots, 16\},$ and we count how many coprime pairs $(a, b)$ with $a < b, \gcd(a, b) = 1$

$$a = 2 \implies b = 3, 5, 7, 9, 11, 13, 15 \implies 7$$ $$a = 3 \implies b = 4, 5, 7, 8, 10, 11, 13, 14, 16 \implies 9$$ $$a = 4 \implies b = 5, 7, 9, 11, 13, 15 \implies 6$$ $$a = 5 \implies b = 6, 7, 8, 9, 11, 12, 13, 14, 16 \implies 9$$ $$a = 6 \implies b = 7, 11, 13 \implies 3$$ $$a = 7 \implies b = 8, 9, 10, 11, 12, 13, 15, 16 \implies 8$$ $$a = 8 \implies b = 9, 11, 13, 15 \implies 4$$ $$a = 9 \implies b = 10, 11, 13, 14, 16 \implies 5$$ $$a = 10 \implies b = 11, 13 \implies 2$$ $$a = 11 \implies b = 12, 13, 14, 15, 16 \implies 5$$

$a = 12 \implies b = 13, 17 \times \implies$ only $13$ is valid $\implies 1$

$$a = 13 \implies b = 14, 15, 16 \implies 3$$ $$a = 14 \implies b = 15 \implies 1$$ $$a = 15 \implies b = 16 \implies 1$$ $$\text{Total} = 7 + 9 + 6 + 9 + 3 + 8 + 4 + 5 + 2 + 5 + 1 + 3 + 1 + 1 = 64$$