A community nutrition educator is organizing a cooking workshop. If the number of participants on two consecutive days are \( p \) and \( q \), and it is known that \( p + q = 150 \), what is the greatest possible value of \(\gcd(p, q)\)?

["Maximizing Humble Ingredients: The Power of GCD in a Cooking Workshop’s Participant Count", "When planning a community event like a cooking workshop, every number matters—not just the headcount, but the hidden mathematical elegance behind it. A local community nutrition educator recently organized a cooking workshop and noticed that on two consecutive days, the number of participants were two positive integers, ( p ) and ( q ), such that ( p + q = 150 ). This simple equation opens a rich opportunity to explore the greatest possible value of (\gcd(p, q))—a measure of their shared foundational connection.", "What is (\gcd(p, q)), and why does it matter in this context? The greatest common divisor reflects the largest number that divides both participant counts evenly. In community planning, aligning such values can foster balance—ensuring resources, recipes, and group activities are shared fairly across all participants.", "Given ( p + q = 150 ), we seek the maximum possible value of (\gcd(p, q)). Let ( d = \gcd(p, q) ). Then we can write:\n[\np = d \cdot a, \quad q = d \cdot b\n]\nwhere ( a ) and ( b ) are positive integers with (\gcd(a, b) = 1) (they are coprime). Substituting into the sum:\n[\nd(a + b) = 150\n]\nThus, ( d ) must be a divisor of 150, and ( a + b = \frac{150}{d} ). Since ( a ) and ( b ) are positive integers and coprime, ( a + b \geq 2 ), so ( \frac{150}{d} \geq 2 ), meaning ( d \leq 75 ).", "To maximize ( d ), we test the largest divisors of 150:\n- Divisors of 150: ( 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150 )", "Try ( d = 75 ):\nThen ( a + b = 150 / 75 = 2 ). The only possibility is ( a = 1, b = 1 ), and (\gcd(1,1) = 1), which satisfies the coprimality condition.", "So, ( p = 75 \cdot 1 = 75 ), ( q = 75 \cdot 1 = 75 ), and indeed ( p + q = 150 ), and (\gcd(75, 75) = 75).", "Any larger ( d ) would require ( a + b < 2 ), which is impossible for positive integers.", "Therefore, the greatest possible value of (\gcd(p, q)) is (\boxed{75}).", "This insight enriches both event design and mathematical understanding—proving that even in cooking and community building, harmony often comes from shared divisibility. Whether feeding minds or meals, knowing the common ground can make all the difference."]