First can I express my delight at the spreadsheet supplied by the Colyton Maths Challenge Group, which enabled them to enter values of A and H to find a solution. I think the cell H8 needed to be "=10*F3+D3" to work (and it does with the right substitution!) - but what a good idea. Well done. Like Angele (no school given), the Colyton group noticed that the problem might be easier with some rearrangement and making JOKE the subject.

At this point it is clear that the maths group, like Andrei (Tudor Vianu College) used exhaustive methods, probably made easier by the spreadsheet.

Here is a alternative approach based on the solution offered by Lee (no school given).

I can rearrange the problem

JOKE = \frac{AHHAAH}{HA}

Now

AHHAAH

is made up of

AH 00 AH

and

HA 00 = 100 \times HA

added together

But $AH 00 AH$ is divisible by $AH$ So $AH 00 AH = 10001 \times AH$

So $AHHAAH = 10001 \times AH + HA \times 100$

$JOKE = \frac{AHHAAH}{HA} = \frac{100 \times HA + AH \times (10001)}{HA}$

The numerator has to be divisible by the denominator (HA) for this to give a four digit answer (JOKE). As $100 \times HA$ is divisble by $HA$ it only needs us to look at $10001 \times AH$

But $10001$ is not prime, it is $73 x 137$

There is no combination of digits such that HA divides AH So $HA$ divides $10001$ giving an answer $73$

$H = 7 A = 3 J = 5 O = 1 K = 6 E = 9$