2008 iTest Problems/Problem 4

Problem

The difference between two prime numbers is $11$. Find their sum.

Solution

We know that any prime number, excluding $2$, is congruent to $1 \pmod 2$. Thus, if both of the primes are not $2$, their difference would be congruent to $0 \pmod 2$. Because $11 \equiv 1 \pmod 2$, one of the primes must be $2$. It follows that the other prime must then be $13$. Therefore, the sum of the two is $13+2=\boxed{15}$.

Solution 2

Since the difference is $11$, one number must be even for the difference to be even. $2$ is the only even prime number, and therefore is one of the two numbers. The other number is $2 + 11 = 13$, and their sum: $13 + 2 = \boxed{15}$.

See Also

2008 iTest (Problems)
Preceded by:
Problem 3
Followed by:
Problem 5
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