Age 16 to 18 Challenge Level:

Find the smallest numbers $a, b$, and $c$ such that: $a^2 = 2 b^3 = 3 c^5.$ What can you say about other solutions to this problem? Congratulations for your good solutions to Ella and Elizabeth, S6, Madras College and Yiwan, The Chinese High Singapore. Here is Yiwan's solution: $$a^2=2b^3=3c^5.$$ As (2,3)=1, that is 2 and 3 have no common divisor other than 1, we shall write $a$, $b$, and $c$ in terms of powers of 2 and 3. Let $c=2^p3^q$ (where p, q are integer numbers above 0). Then $$3c^5=2^{5p}3^{5q+1}=2b^3=a^2.$$ Hence $$b=2^{(5p-1)/3}3^{(5q+1)/3};$$ $$a=2^{5p/2}3^{(5q+1)/2}.$$ As a, b are all integers, it follows that $3| (5p-1)$, $3| (5q+1)$, $2| (5p)$ and $2| (5q+1)$ [using the notation $3| (5p-1)$ to mean 3 divides or is a factor of $(5p-1)$]. Obviously the solution for the smallest number is when p=2 and q=1. In this case, $c=2^2 \times 3=12$ ; $b=2^3 \times 3^2=72$ ; $a=2^5 \times 3^3=864.$ The smallest solution is $(a=864, b=72, c=12).$ For other solutions take $$p=\{2, 8, 14, 20, ... 6m + 2\}$$ where m is a positive integer and $$q=\{1, 7, 13, 19, ... 6n - 5\}$$ where n is a positive integer. If we substitute any value of $p$ and $q$ from the corresponding domain, we will get the other solutions for the equation.