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Mathematics
Problem
Let
S
k
=
1
+
2
+
⋯
+
k
k
S_k = \frac{1+2+\dots+k}{k}
S
k
=
k
1
+
2
+
⋯
+
k
and
∑
j
=
1
n
S
j
2
=
n
A
(
B
n
2
+
C
n
+
D
)
\sum_{j=1}^{n} S_j^2 = \frac{n}{A}(Bn^2 + Cn + D)
∑
j
=
1
n
S
j
2
=
A
n
(
B
n
2
+
C
n
+
D
)
, where
A
,
B
,
C
,
D
∈
N
A, B, C, D \in \mathbb{N}
A
,
B
,
C
,
D
∈
N
and A has least value. Then [Main April 8, 2023 (I)] (a)
A
+
B
A + B
A
+
B
is divisible by
D
D
D
(b)
A
+
B
=
5
(
D
−
C
)
A + B = 5(D - C)
A
+
B
=
5
(
D
−
C
)
(c)
A
+
C
+
D
A + C + D
A
+
C
+
D
is not divisible by
B
B
B
(d)
A
+
B
+
C
+
D
A + B + C + D
A
+
B
+
C
+
D
is divisible by 5
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