1) Find the number of zero's in the expansion of
Consider the number: .
The question becomes: how many 5's are in the factorization?
(Since "5 times an even number" produces a 0.)
Every fifth number has a 5: .
. . There are fifteen 5's among the factors.
But a few of them two 5's . . . these are the multiples fo 25.
. . There are: . of them.
Each of these contribute an addition 5 to the factoring.
Hence, there are: . factors of 5.
. . Each of them, paired with a factor 2, produces a 10.
Therefore, has eighteen 0's at the end.
As we have seen, 79! ends in 18 zeros.2) Find the last two digits of
Multiplying by 80, 81, 82, 83, ... 95, we get even more zeros at the end.
Obviously, the last two digits are 00.