Three numbers are pairwise co-prime (each pair has greatest common divisor 1). The product of the first two numbers is 551 and the product of the last two numbers is 1073. What is the sum of the three numbers?

Introduction / Context:
This is a number theory problem involving co-prime numbers and factorisation. You are given that three numbers are pairwise co-prime, meaning no two share any common factor greater than 1. You are also given the products of the first two and the last two numbers. From this information, you must determine the individual numbers and then find their sum. This type of question requires careful use of prime factorisation and the co-prime condition.

Given Data / Assumptions:

• Let the three numbers be a, b and c.
• The product of the first two is a * b = 551.
• The product of the last two is b * c = 1073.
• The numbers are pairwise co-prime, so any two of them have greatest common divisor 1.
• We must find a + b + c.

Concept / Approach:
First factorise the given products into primes. Then use the pairwise co-prime condition to identify which prime factors belong to which number. Since a and c are both multiplied with b in the given products, any common factor between 551 and 1073 must be part of b. Once we identify b, we can find a and c by dividing the products by b. Finally, we verify the co-prime condition for all pairs and add the three numbers to obtain the required sum.

Step-by-Step Solution:
Step 1: Factorise 551. Try dividing by small primes: 551 is not even and digit sum 5 + 5 + 1 = 11 is not divisible by 3. Check 19: 19 * 29 = 551. So 551 = 19 * 29. Step 2: Factorise 1073. It is not divisible by 2, 3 or 5. Check 29: 29 * 37 = 1073. So 1073 = 29 * 37. Step 3: The common factor between 551 and 1073 is 29. Because a, b, c are pairwise co-prime, any common factor between the products a * b and b * c must belong to b. Step 4: Therefore, b must be 29. Step 5: From a * b = 551, we get a = 551 / 29 = 19. Step 6: From b * c = 1073, we get c = 1073 / 29 = 37.

Verification / Alternative check:
Check the pairwise co-prime condition. gcd(19, 29) = 1, gcd(19, 37) = 1 and gcd(29, 37) = 1, so the three numbers are indeed pairwise co-prime. Also verify the products: 19 * 29 = 551 and 29 * 37 = 1073, which match the given information. Now compute the sum: a + b + c = 19 + 29 + 37 = 85. Everything is consistent, confirming that the answer is correct.

Why Other Options Are Wrong:
Option 83, 89, 79 and 91: These sums would correspond to different sets of numbers. If you try to construct numbers that are pairwise co-prime and whose products match 551 and 1073, you will find that only 19, 29 and 37 work. Any other combination produces a different product or violates the co-prime condition.

Common Pitfalls:
A common error is to misinterpret “co-prime to each other” and assume only that the three numbers share no common divisor all together, instead of checking each pair separately. Another mistake is incomplete factorisation of 551 or 1073, which leads to wrong candidates for the three numbers. Always factor completely into primes and then distribute the prime factors according to the co-prime requirement.

Final Answer:
The sum of the three pairwise co-prime numbers is 85.