Prime Fun!

Summing the two equations: $2a=73$ $a=37$

Therefore, $b+c=15$ .

The sum of an odd number and an even number is odd: one of b and c must be odd and the other even.

There is only one even prime: 2 (which is also the smallest). Therefore, $c=2$ .

From that, $b=13$ .

Finally, $abc = 37 \times 13 \times 2 = \boxed{962}$

37+13+2=52 37-13-2=22 now 37 13 2=962

(a+b+c) - (a-b-c) = 52-22 ---> 2b + 2c = 30 --> 2(b+c) = 30 ----> b+c = 15 ----> (a+b+c) - (b+c) = 52 - 15 ---> a= 37. Because b and c are prime numbers so 15 = 13 + 2 ---> b>c so b= 13 and c = 2 . Thus abc = 37.13.2 = 962

a+(b+c)=52 a-(b+c)=22 a=37 b+c=15 hence b=13 and c=2

You can shorten your prime number search by realizing that $a- ( 52 - a ) = 22$ . By starting at $47$ and working down the primes, you quickly realize that $a = 37$ , and thus $b = 13$ and $c = 2$ .

$37 \times 13 \times 2 = 962$