A9. The number 3600 can be written as $2^{a} \times 3^{b} \times 4^{c} \times 5^{d}$, where $a, b, c$ and $d$ are all positive integers. It is given that $a+b+c+d=7$.
What is the value of $c$ ?
Solution
1

First note that $3600=2^{4} \times 3^{2} \times 5^{2}$. We require $3600=2^{a} \times 3^{b} \times 4^{c} \times 5^{d}$. Comparing the two expressions gives $b=2, d=2$ and $2^{4}=2^{a} \times 4^{c}$. Since $4^{c}=2^{2 c}$, we have $a+2 c=4$ and so $a=4-2 c$. Also, since $a+b+c+d=7$, we have $4-2 c+2+c+2=7$ which has solution $c=1$.