We have as 31 prime using Wilsons theorem
30! + 1 \equiv 0 \pmod {31}
or 30 * 29! + 1 \equiv 0 \pmod {31}
as 30 \equiv - 1 \pmod {31}
so we have -1 * 29! + 1 \equiv 0 \pmod {31}
Or 29! -1 \equiv 0 \pmod {31}
Or 4 * 29! - 4 \equiv 0 \pmod {31}
adding 124 which is multiple of 31 we get
4 * 29! + 120 \equiv 0 \pmod {31}
or 4 * 29! + 5! \equiv 0 \pmod {31}
Proved
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