find the remainder when 1!+2!+3!+..........99!+100! divided by 12.

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- Apr 18th 2008, 07:49 PMmythri sharmaplz help to solve
find the remainder when 1!+2!+3!+..........99!+100! divided by 12.

- Apr 18th 2008, 07:55 PMMathstud28
- Apr 18th 2008, 08:34 PMawkward
- Apr 19th 2008, 01:28 AMIsomorphism
Try writing the factorials as a product.

Now**12 = 4 x 3**

1! = 1

2! = 2 x 1

3! = 3 x 2 x 1

4! = (4 x 3) x 2 x 1 <------ There is a 4 x 3 here

5! = 5 x (4 x 3) x 2 x 1 <------ There is a 4 x 3 here

6! = 6 x 5 x (4 x 3) x 2 x 1 <------ There is a 4 x 3 here

7! = 7 x 6 x 5 x (4 x 3) x 2 x 1 <------ There is a 4 x 3 here

Well I am sure you can see by now that 12 will divide any factorial greater than or equal to 4. That means the only fellows who can leave remainders are 1!+2!+3! = 1+2+6 = 9. So 9 is the remainder when divided by 12.

Now can you try a related question:

"Which is the largest factorial number that is not divisible by 27?"

Try it :D - Apr 19th 2008, 04:35 PMMathstud28
- Apr 20th 2008, 06:10 AMIsomorphism
- Apr 20th 2008, 06:35 AMMathstud28