(a) What is the largest integer $n$ such that $7^n$ divides $500!$ ?

Here's my attempt at (a).....

Notice that .....

500/7  = 71 numbers that have at least one factor of 7

500/49  = 10 more numbers that have an additional  factor of 7

500/343 = 1 more number that adds an additional factor of 7

So.....71 + 10 + 1   = 82

So ..... 7^82  will divide 500!....thus, n = 82

That is really good logic I think Chris    

I forgot the 'divide by 49' one when i did it.                

Here's (b)....it's pretty much an extension of (a)

1000/7 = 142 numbers that have at least 1 factor of 7

1000/49 = 20 numbers that have at  an additional factor of 7

1000/343 =  2 numbers that have a third factor of 7

So.........    142 + 20 + 1 =  163

And 7^163 will divide 1000!   ...so n = 163

Does anyone know how to work out problem (c)?

(c) Based on the other two, (a) and (b), we know that there are 164 powers of 7 that divide into 1000!, and there are 82 factors of 7 that divide into 500!. Since we are dividing the same numbers that have different powers, we subtract the denominator's exponent (82) from the numerator's exponent (164). However, the denominator is SQUARED, meaning that we subtract 82 TWICE from 164.

$164-82-82=0$

To put it mathematically, all the powers of 7 are canceled, and that leaves 7^0 = 1. N, the exponent of 7, is 0.