In this series, each term which is greater than or equal to the 5th, 5!, includes 5 in its factorial.
Therefore, each term is divisible by 5, so that the sum of them is divisible by 5 as well.
That is: 5!+6!+ … +150!+151! ≡ 0 (mod 5) ··· Ex.1
While, 1!+2!+3!+4!=33=5x6+3 so, the sum of 1!+2!+3!+4! is not divisible by 5.
That is: 1!+2!+3!+4! ≡ 3 (mod 5) ··· Ex.2
Add Ex.1 and Ex.2. We have: 1!+2!+3!+4!+ …+150!+151! ≡ 3 (mod 5)
Therefore, the given series is not divisible by 5. If the series is divided by 5, the remainder is 3.