Three positive integers are pairwise coprime. The product of the first two is 437, and the product of the last two is 551. Find the sum of all three integers.

Introduction / Context:
This problem blends factorization with the concept of pairwise coprimality. By comparing two products that share one variable, we can isolate each integer using greatest common divisors (gcds) and prime factorizations.

Given Data / Assumptions:

• a, b, c are pairwise coprime (any two have gcd 1).
• a*b = 437
• b*c = 551
• All are positive integers.

Concept / Approach:
Since a and c are coprime (pairwise condition), the only common factor between ab and bc is b itself. Therefore gcd(437, 551) equals b. Factor the numbers to compute the gcd and then find a and c by division.

Step-by-Step Solution:
Prime factorization: 437 = 19*23; 551 = 19*29.gcd(437, 551) = 19 ⇒ b = 19.Then a = 437 / 19 = 23; c = 551 / 19 = 29.Sum = a + b + c = 23 + 19 + 29 = 71.

Verification / Alternative check:
Check pairwise coprime: gcd(23, 19) = 1, gcd(19, 29) = 1, gcd(23, 29) = 1. Products match the given values. Everything is consistent.

Why Other Options Are Wrong:
91, 81, 70, and 73 result from incorrect gcd identification or misfactorization, leading to wrong values for a, b, or c.

Common Pitfalls:
Overlooking pairwise coprime condition; assuming common factors between a and c; arithmetic slips in factorization or division.

Final Answer:
71