You may also like

Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

Ladder and Cube

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Bendy Quad

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

Square LCM

Age 14 to 16 Short
Challenge Level

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)

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.