The remeainder obtained when 1! + 2! + 3! + … + 77! is divided by 7 is :

Options

A. 0
B. 5
C. 4
D. can't be detemined

Answer

To find the remainder when \( 1! + 2! + 3! + \ldots + 77! \) is divided by 7, we can analyze the factorials step by step.

### Step 1: Calculate the factorials for small numbers
We start by calculating the factorials of the first few integers:

- \( 1! = 1 \)
- \( 2! = 2 \)
- \( 3! = 6 \)
- \( 4! = 24 \)
- \( 5! = 120 \)
- \( 6! = 720 \)
- \( 7! = 5040 \)

### Step 2: Find the remainders when divided by 7
Next, we find the remainders of these factorials when divided by 7:

- \( 1! \mod 7 = 1 \)
- \( 2! \mod 7 = 2 \)
- \( 3! \mod 7 = 6 \)
- \( 4! \mod 7 = 24 \mod 7 = 3 \)
- \( 5! \mod 7 = 120 \mod 7 = 1 \)
- \( 6! \mod 7 = 720 \mod 7 = 6 \)
- \( 7! \mod 7 = 0 \) (and all higher factorials will also be 0)

### Step 3: Sum the remainders
Now we sum the remainders we calculated:

\[ 1 + 2 + 6 + 3 + 1 + 6 + 0 + 0 + \ldots + 0 = 19 \]

### Step 4: Find the remainder of the sum when divided by 7
Finally, we find the remainder of the sum \( 19 \) when divided by \( 7 \):

\[ 19 \mod 7 = 5 \]

### Conclusion
Thus, the remainder obtained when \( 1! + 2! + 3! + \ldots + 77! \) is divided by \( 7 \) is \( 5 \).

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