MathDB
Problems
Contests
National and Regional Contests
India Contests
India LIMIT
2019 LIMIT
2019 LIMIT Category B
Problem 2
Problem 2
Part of
2019 LIMIT Category B
Problems
(2)
units digit of factorial sum
Source: LIMIT 2019 CAS2 P7
4/28/2021
The digit in unit place of
1
!
+
2
!
+
…
+
99
!
1!+2!+\ldots+99!
1
!
+
2
!
+
…
+
99
!
is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
3
<span class='latex-bold'>(A)</span>~3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
0
<span class='latex-bold'>(B)</span>~0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
1
<span class='latex-bold'>(C)</span>~1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
7
<span class='latex-bold'>(D)</span>~7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
7
factorial
number theory
complex subsets
Source: LIMIT 2019 CBS2 P2
4/28/2021
Let
C
\mathbb C
C
denote the set of all complex numbers. Define
A
=
{
(
z
,
w
)
∣
z
,
w
∈
C
and
∣
z
∣
=
∣
w
∣
}
A=\{(z,w)|z,w\in\mathbb C\text{ and }|z|=|w|\}
A
=
{(
z
,
w
)
∣
z
,
w
∈
C
and
∣
z
∣
=
∣
w
∣
}
B
=
{
(
z
,
w
)
∣
z
,
w
∈
C
and
z
2
=
w
2
}
B=\{(z,w)|z,w\in\mathbb C\text{ and }z^2=w^2\}
B
=
{(
z
,
w
)
∣
z
,
w
∈
C
and
z
2
=
w
2
}
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
A
=
B
<span class='latex-bold'>(A)</span>~A=B
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
A
=
B
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
A
⊂
B
and
A
≠
B
<span class='latex-bold'>(B)</span>~A\subset B\text{ and }A\ne B
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
A
⊂
B
and
A
=
B
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
B
⊂
A
and
B
≠
A
<span class='latex-bold'>(C)</span>~B\subset A\text{ and }B\ne A
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
B
⊂
A
and
B
=
A
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
None of the above
<span class='latex-bold'>(D)</span>~\text{None of the above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
None of the above
complex numbers
algebra