Triangular inequality:
The sum of any two sides of a triangle
must be greater than the third side.
The primes under 50 are
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47
We obviously cannot use 2 for the smallest side,
for 2 plus any larger prime cannot be greater than
the next prime.
Try 3 for the smallest side:
3+5=8 and the only prime the third side could be is 7,
but the perimeter would be 3+5+7=15, which is not prime
If we used 7 for the middle-sized side, then 3+7=10,
leaving no possibility for the largest side.
If we used 11 for the middle-sized side, then 3+11=14,
which leaves no possibility for the largest side.
If we used 13 for the middle-sized side, then 3+13=16,
and the 3rd side could only be 13, but the triangle
would be isosceles, and besides, the perimeter would
not be prime.
If we used 17 for the middle-sized side, then 3+17=20,
and the 3rd side could only be 19, but the triangle
the perimeter would be 39 which is not prime, and
besides, it's larger than any of the choices.
If we used anything larger for the middle-sized side,
we would get something larger than any of the choices listed.
So we've ruled out 3 as a possibility for the smallest side.
So we try 5 as the smallest side.
Then we try 7 for the middle size side, 5+7=12, so
the largest side could only be 11, and the perimeter would be
5+7+11 = 23, and eureka! 23 is prime!
So 23 has to be the smallest prime perimeter!
choice e)
The perimeter is the prime number 23 when the sides are
prime numbers 5, 7, and 11.
Edwin
