MathNet 2026 — Part 03 Question 5

a) Notice the sums:

Check it's worth:
$$
\begin{gathered}
\frac{(k+1) \cdot (k+2) \cdot (k+3) \cdot \ldots \cdot (k+901)}{901} + (k+2) \cdot (k+3) \cdot (k+4) \cdot \ldots \cdot (k+901) = \\
\frac{(k+2) \cdot (k+3) \cdot \ldots \cdot (k+901) \cdot (k+902)}{901}
\end{gathered}
$$


b) Be $N$ The sum of the numbers:
$$
\begin{array}{rrrrrrrrrr}
& 1 & \times & 2 & \times & 3 & \times & \ldots & \times & 900 \\
+ & 2 & \times & 3 & \times & 4 & \times & \ldots & \times & 901 \\
+ & \ldots & & & & & & & & \\
+ & 1116 & \times & 1117 & \times & 1118 & \times & \ldots & \times & 2015
\end{array}
$$
show that $901 \cdot N$ is divisible by every element of the set $\{1116,1117, \ldots, 2016\}$.