Factors and Factorials question

[sezuno]

Hi,

 

I'm a new member and was hoping to get some help on the following question:

 

If 15!/3^m is an integer, what is the greatest possible value of M?

 

a. 4

b. 5

c. 6

d. 7

e. 8

 

I know the answer is c, but how?

Thanks.

[800needed]

Welcoem to TM! :) Anyway the perfet approach to this question is to write out 15!

 

So 15!=15*14*13*12*11*10*9*8*7*6*5*4*3*2*1

 

So out of thoe what numbers are divisible by 3? well 15, 12, 9, 6, and 3.

 

so 15=5*3, 12=3*4, 9=3*3, 6=3*2, and 3=3*1

 

Now count how many 3 you have there and you will see you have six 3s. so the answer can only be 6. Hope this help!:tup:

[udaygrover]

Absolutely right...!!

 

Now lets look at the question in a diferent way....

What if the question said 100!/2^m

 

One cannot sit and find all numbers which are prsent and then add the nuber of 2's in it..right...so here is the trick

 

first we need to find the number of multiples of 2

== 50

 

Then the number of multiples of 4

number of multiples of 8

number of multiples of 16

number of multiples of 32

number of multiples of 64

 

Therfore the answer for maximum value of m is == 50+25+12+6+3+1 = 97

 

if we do this for the current question

 

15!/3^m

 

number of factors of 3

number of factors of 9

answer == 5 + 1 = 6

 

Hope u get the answer

 

please pm me if u have ne more doubts on the same

 

regards

uday

[Stormgal]

Absolutely right...!!

 

Now lets look at the question in a diferent way....

What if the question said 100!/2^m

 

One cannot sit and find all numbers which are prsent and then add the nuber of 2's in it..right...so here is the trick

 

first we need to find the number of multiples of 2

== 50

 

Then the number of multiples of 4

number of multiples of 8

number of multiples of 16

number of multiples of 32

number of multiples of 64

 

Therfore the answer for maximum value of m is == 50+25+12+6+3+1 = 97

 

if we do this for the current question

 

15!/3^m

 

number of factors of 3

number of factors of 9

answer == 5 + 1 = 6

 

Hope u get the answer

 

please pm me if u have ne more doubts on the same

 

regards

uday

 

Hi Uday - you are a genius - I was about to ask that question, that what if the number is large.

Anyway, I hate to sound dumb, but why would one want to find the multiples of not only 2, but of 4, 8, 16, 32, etc?

[Vakul]

Hi Uday - you are a genius - I was about to ask that question, that what if the number is large.

Anyway, I hate to sound dumb, but why would one want to find the multiples of not only 2, but of 4, 8, 16, 32, etc?

 

multiple of 2 will have only 1 factor of 2.

multiple of 4 will have 2 factors of 2

multiple of 8 will have 3 factors of 2

multiple of 16 will have 4 factors of 2

[udaygrover]

Hi...

 

Thats right...

 

Hope you have understood the reason strom

 

let me know if you still have a doubt

 

uday

[texaspunk]

Thanks all for excellent explanation. My only clarification is that the explanation states:

 

find multiples of 2

and

find multiples of 4

 

using the "less than symbol makes this explanation is a bit confusing.

 

Because as we know the number of multiples of 2 must include 100 in order to get an answer of 25 and and the number of multiples of 4 in 100 must include the number 100 in order for the answer to be 25. (not

or

Am I missing something?:mad: