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Byju's Answer
Standard XII
Mathematics
Factorial
Find the high...
Question
Find the highest power of 5 dividing 100! and find number of zero at the end of 100!
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Solution
Highest exponent of a prime number a in n! is
=
[
n
a
]
+
[
n
a
2
]
+
[
n
a
3
]
+
.
.
.
.
where [.] is greatest integer formation
⇒
Exponent of
5
in
100
!
is
=
[
100
5
]
+
[
100
5
2
]
+
[
100
5
3
]
+
.
.
.
.
=
20
+
4
+
0
+
0
.
.
.
.
=
24
To make a zero in the end, use need a 10 as a factor
i.e., a product of
2
and
5
Exponent of 2 in
100
!
will be more than that of
5
, as
2
is smaller number.
Hence
5
24
will multiply with
2
24
to make
10
24
⇒
24
zeroes are there at the end of
100
!
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