Question

Write remainder when 1! + 2! + 3! +. . . + 175! is divided by 15.

Answer 1

when
1! divided by 15 remainder is 1
2! divided by 15 remainder is 2
3! divided by 15 remainder is 6
4! divided by 15 remainder is 9
5! divided by 15 remainder is 0
6! divided by 15 remainder is 0..
so only 1!+2!+3!+4! is to be considered
1+2+6+24=33
33/15= remainder is 3

Answer 2

Concept

Remainder, as its name suggests, is something that "remains" after a task is completed.

Remainder refers to the remaining portion after the division process is complete.

Given

It is given that 1! +2! + 3! +. . . + 175! is divided by 15.

Find

We need to find the remainder

Solution

5!=1⋅2⋅3⋅4⋅5 is divisible by 15

Therefore for any number n≥5 we have n!=1⋅2⋅3⋅4⋅5⋯n is divisible by 15.

Therefore, the remainder when 1!+2!+⋯+175! is divided by 15 is the same as the remainder when 15 divides 1!+2!+3!+4!.

Now 1!+2!+3!+4!=1+2+6+24=33 is divided by 15, the remainder is 3.

Hence the remainder is 3

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