A number theory problem by Christopher Boo

For any positive integer nn, the following is true:

nn is divisible by 1.1.
n(n+1)n(n+1) is divisible by 1×2.1\times 2.
n(n+1)(n+2)n(n+1)(n+2) is divisible by 1×2×3.1\times 2\times 3.

Is it true that n(n+1)(n+2)(n+3)n(n+1)(n+2)(n+3) must be divisible by 1×2×3×4?1\times 2 \times 3\times 4?

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