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 $113-13=\boxed{\textbf{(B) } 100}$ .

1973 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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1973 AHSME Problems/Problem 3

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