Square LCM
Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?
Problem
The highest common factor of two positive integers $m$ and $n$ is $12$, and their lowest common multiple is a square number.
How many of the five numbers $\frac{n}{3}$, $\frac{m}{3}$, $\frac{n}{4}$, $\frac{m}{4}$ and $mn$ are square numbers?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: two of them
$m$ and $n$ are multiples of $12$, so say $m=12a$ and $n=12b$
$12$ the highest common factor $\therefore$ $a$ and $b$ don't have any common factors
Lowest common multiple of $12a$ and $12b$ is $12ab$ (since $a,b$ have no common factors)
$12ab$ is a square number so it is a product of squares
$12ab=4\times3\times a \times b$
Say $3a$ is a square, so $a=3p^2$ for some number $p$ (so that $3a=9p^2=(3p)^2$)
and $b=q^2$
So $m$ and $n$ are $36p^2$ and $12q^2$
$\Rightarrow \frac m3$ and $\frac n3$ are $12p^2$ (not square), $4q^2$ (square)
$ \frac m4$ and $\frac n4$ are $9p^2$ (square), $3q^2$ (not square)
$mn = 12\times36\times p^2\times q^2$ (not square)