What is the largest integer n such that 3^{n} is a factor of 1 x 3 x 5 x... x 97 x 99?

Guest Feb 16, 2020

#2**+1 **

What is the largest integer n such that 3^n is a factor of 1 x 3 x 5 x... x 97 x 99?

How many of those numbers has a factor of 3

3,6,9, ........99 There are 33 of them

Divide each of these by 3

1,2,3,...... 33

Which of these have a factor of 3

3,6,9, ... 33 There are 11 of them

Divide each of these by 3

1,2,3,...... 11

Which of these have a factor of 3

3,6,9 There are 3 of them

Divide each of these by 3

1,2,3

Which of these have a factor of 3

3 There are 1 of them

33+11+3+1= 48

**I think that the largest n is 48**

Melody Feb 16, 2020

#5**+3 **

See https://web2.0calc.com/questions/help_40754#r2

We have yet another annoying situation where someone posts the question twice instead of continuing the original post!

Alan Feb 16, 2020

#8**+1 **

**Thanks Alan **

Ok, WolframAlpha and Alan both agree that the answer is 26 so where did I go wrong...

I did not account for the even numbers to be missing.

I found the power if all the numbers from 1 to 99 were multiplied together.

Silly me,

I will try my way again,

3,9,15,21,27,33,39,45,51,57,63,69,75,81,87, 93,99 are all divisible by 3 (That is 17 numbers)

divide by 3

1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33

which of these are divisible by 3

3,9,15,21,27,33 (That is 6 numbers)

Divide by 3

1,3,5,7,9,11

which of these are divisible by 3

3,9, (That is 2 numbers)

Divide by 3

1,3

which of these are divisible by 3

3, (That is 1 number)

17+6+2+1 = 26

Now we are in agreement

Melody Feb 16, 2020