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 =

AHHAAH

Now AHHAAH is made up of AH00AH and HA00 = 100 ×HA added together

But AH00AH is divisible by AH So AH00AH = 10001 ×AH

So AHHAAH = 10001 ×AH + HA ×100

JOKE = AHHAAH
HA
= 100 ×HA +AH ×(10001)
HA

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

But 10001 is not prime, it is 73x137

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