# 1973 AHSME Problems/Problem 3

## Problem

The stronger Goldbach conjecture states that any even integer greater than 7 can be written as the sum of two different prime numbers. For such representations of the even number 126, the largest possible difference between the two primes is $\textbf{(A)}\ 112\qquad\textbf{(B)}\ 100\qquad\textbf{(C)}\ 92\qquad\textbf{(D)}\ 88\qquad\textbf{(E)}\ 80$

## Solutions

We can guess and check small primes, subtract it from $126$, and see if the result is a prime because the further away the two numbers are, the greater the difference will be. Since $126 = 2 \cdot 3^2 \cdot 7$, we can eliminate $2$, $3$, and $7$ as an option because subtracting these would result in a composite number.

If we subtract $5$, then the resulting number is $121$, which is not prime. If we subtract $11$, then the resulting number is $115$, which is also not prime. But when we subtract $13$, the resulting number is $113$, a prime number. The largest possible difference is $113-13=\boxed{\textbf{(B) } 100}$.