MathDB
Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 8
2017 AMC 8
19
19
Part of
2017 AMC 8
Problems
(1)
To Factor Wouldn't Hurt
Source: AMC 8 2017 #19
11/22/2017
For any positive integer
M
M
M
, the notation
M
!
M!
M
!
denotes the product of the integers
1
1
1
through
M
M
M
. What is the largest integer
n
n
n
for which
5
n
5^n
5
n
is a factor of the sum
98
!
+
99
!
+
100
!
98!+99!+100!
98
!
+
99
!
+
100
!
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
23
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
24
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
25
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
26
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
27
<span class='latex-bold'>(A) </span>23\qquad<span class='latex-bold'>(B) </span>24\qquad<span class='latex-bold'>(C) </span>25\qquad<span class='latex-bold'>(D) </span>26\qquad<span class='latex-bold'>(E) </span>27
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
23
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
24
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
25
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
26
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
27
AMC 8