Question

F 1! + 2! + 3! + . + 50! Is divided by 5, then the remainder is

Answer 1

Solution :-

◾First of all we should know what do we mean by n! or n factorial.

⏩The consecutive multiplication of "n" natural numbers is termed as n! or n factorial.

eg. 6! = 1 × 2 × 3 × 4 × 5 × 6

◾Now solving the question :-

◾The reminder when

◾The reminder when 1! + 2! + 3! + 4! + 5! ....... +50!

◾The reminder when 1! + 2! + 3! + 4! + 5! ....... +50! is divided by 5

◾As we know that

→ n! = 1 × 2 × 3 ...... (n-2)(n-1)(n).

◾Then from this we can say

5 | 5! ( 5 divides 5! )

as 5! = 1 × 2 × 3 × 4 × 5

◾Also

5 | 6! (5 divides 6!)

as 6 ! = 1 × 2 × 3 × 4 × 5 × 6

or

→ 6! = 5! × 6

◾So we can conclude that every factorial after 5! including itself is divisible by 5

▪️Then

→ 1! + 2! + 3! + 4! + 5!( 1 + 6 ....+ (50! ÷ 5!))

let

→ 1 + 6 ....+ (50! ÷ 5!) = k

▪️Then

→ 1! + 2! + 3! + 4! + 5k

◾So the reminder will only be obtained from

1! + 2! + 3! + 4!

▪️Now sum of above :-

= 1 + 2 + 6 + 24

= 1 + 2 + 30

◾As 30 is divisible by 5 hence the reminder obtained

= 1 + 2

= 3 .