BdMO Problem 6

Use casework.

The required age equation can be written as $2x+y+z=23$ . The twins' combined age (i.e. $2x$ ) is an even number on the LHS, which requires $y$ to be even (odd) and $z$ to be odd (even). WLOG, let $z = 2$ be one of the prime ages in order to obtain $2x+y=21$ (with $x = 2,3,5,7$ ).

If $x = 2, 7$ , then we would have triplets $\Rightarrow$ contradiction.

If $x=3$ , then $y = 15$ , which is composite.

If $x=5$ , then $y=11 \Rightarrow$ acceptable.

Finally, we have $(x,y,z) = (5,11,2) \Rightarrow 2x = \boxed{10}.$

Sum of the ages of twins is 10. 5 is a prime number. 5*2=10; The other two numbers are 11 and 2, since both these are also prime numbers. age of eldest brother = 11 years. age of 1st twin = 5 years. age of 2nd twin = 5 years. age of youngest brother = 2 years. 11+5+5+2=23.

try and error method

first primes are 2,3,5,7 either 2or 3 doesn't satisfy the rule.hence answer is 5