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MAA AMC
AMC 8
2022 AMC 8 -
17
17
Part of
2022 AMC 8 -
Problems
(1)
There's Even A Double Factorial!! (2022 AMC 8 P17)
Source:
1/25/2022
If
n
n
n
is an even positive integer, the double factorial notation
n
!
!
n!!
n
!!
represents the product of all the even integers from
2
2
2
to
n
n
n
. For example,
8
!
!
=
2
⋅
4
⋅
6
⋅
8
8!! = 2 \cdot 4 \cdot 6 \cdot 8
8
!!
=
2
⋅
4
⋅
6
⋅
8
. What is the units digit of the following sum?
2
!
!
+
4
!
!
+
6
!
!
+
⋯
+
2018
!
!
+
2020
!
!
+
2022
!
!
2!! + 4!! + 6!! + \cdots + 2018!! + 2020!! + 2022!!
2
!!
+
4
!!
+
6
!!
+
⋯
+
2018
!!
+
2020
!!
+
2022
!!
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
8
<span class='latex-bold'>(A)</span> ~0\qquad<span class='latex-bold'>(B)</span> ~2\qquad<span class='latex-bold'>(C)</span> ~4\qquad<span class='latex-bold'>(D)</span> ~6\qquad<span class='latex-bold'>(E)</span> ~8\qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
8
factorial
AMC 8
2022 AMC 8