Question

What is the remainder left after dividing \[1! + 2! + 3! + ...... + 100!\] by 7?
A) 0
B) 5
C) 21
D) 14

Answer 1

Hint:
Here we will first find the terms of the series which are divisible by number 7. Then we will find the sum of the remaining terms of the series. Then we will divide the sum by the number 7 to get the value of the remainder.

Complete Step by step Solution:
Given series is \[1! + 2! + 3! + ...... + 100!\].
Now, we will find those factorial number terms in the given series which is divisible by 7. Therefore, we get
\[7! + 8! + 9! + ...... + 100!\] are the numbers in the series which is divisible by 7.
So the remaining terms are \[1! + 2! + 3! + 4! + 5! + 6!\].
Now we will find the sum of the remaining series number and divide it by number 7 to get the value of the remainder.
So the remaining series is \[1! + 2! + 3! + 4! + 5! + 6!\]
Now we will find the value of this series. Therefore, we get
\[ \Rightarrow 1! + 2! + 3! + 4! + 5! + 6! = 1 + 2 + 6 + 24 + 120 + 720\]
\[ \Rightarrow 1! + 2! + 3! + 4! + 5! + 6! = 873\]
Now we will divide this by number 7 to get the value of the remainder.
So, when we divide 873 by the number 7 we will get the value of the remainder as 5.
Hence the remainder left after dividing \[1! + 2! + 3! + ...... + 100!\] by 7 is 5.

So, option B is the correct option.

Note:
We should know that the division is the operation in which the dividend is divided by the divisor to get the quotient along with some remainder. So, the general formula of the division operation is
Dividend \[ =\] (Divisor $\times $ Quotient) \[+\] Remainder
The dividend is the term or number which is to be divided. The divisor is the term or number which we divide the dividend. The quotient is the term or number which is the answer of the division operation and the remainder is the term which is left when a division operation is performed.