Question 574892
If she divides the cookies evenly among 7 plates, 4 cookies are left.
<pre>
That means it must be 4 more than a multiple of 7.

The multiples of 7 that we can add 4 to and get a result between 24 
and 59 are

21, 28, 35, 42, 49

4 more than those are

25, 32, 39, 46, 53

So it has to be one of those five numbers.
</pre>
If she divides the cookies evenly among 6 plates, 5 cookies are left.
<pre>
That means it must be 5 more than a multiple of 6

So we will subtract 5 from each of these 

25, 32, 39, 46, 53

to see which one gives us a multiple of 6.

20, 27, 34, 41, 48

The only one of those which is a multiple of 6 is 48, which 
is what we got when we subtracted 5 from 53.

So the answer is 53.

Edwin</pre>