Product of $N$ Consecutive Integers Divided by $N$

Number Theory Level 1

1 is divisible by 1.
$2\times 3$ is divisible by 2.
$4\times5\times6$ is divisible by 3.
$7 \times 8 \times 9 \times 10$ is divisible by 4.
$11 \times 12 \times 13 \times 14 \times 15$ is divisible by 5.

True or False?

The product of any $N$ consecutive integers must be divisible by $N.$

Product of NNN Consecutive Integers Divided by NNN

Product of $N$ Consecutive Integers Divided by $N$