Square LCM
Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?
Age
14 to 16
Challenge level
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.
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)