Greatest Value of K

[satyavegi]

Q. X is the product of each integer from 1 to 50 inclusive and y=100^k,where k is an integer
What is the greatest value of K for which Y is a factor of X

A)0
B)5
c)6
d)10
e)12

OA C

[Anurag@Gurome]

Q. X is the product of each integer from 1 to 50 inclusive and y=100^k,where k is an integer What is the greatest value of K for which Y is a factor of X

We need to calculate the number of zeros at the end of X = 50!

To get one zero at the end of a number we a pair of 2 and 5 in its factors. As number of 5s are less than number of 2s, let us calculate number of 5s in X.

The sources of 5s in X are 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50. Each of 5, 10, 15, 20, 30, 35, 40, and 45 will contribute one 5 and 25 and 50 will contribute one 5s each.

Hence, total number of 5s as factor in X = (1*8 + 2*2) = 12

Hence, there will be 12 zeros at the end of X.
Hence, number of times X can be completely divided by 100 is (12/2) = 6

Therefore, k = 6

The correct answer is C.

[1947]

since Y is a factor of X
it means X=N.Y
i.e. 50!=N.(100^K)
50!=N.(5^2.2^2)^k
since there are many 2 in left hand side we just need to find number of 5's
There are 10 multiples of 5 of which 25 and 50 have two 2's so 12 in total
so k=12/2
=6

[satyavegi]

Hi Anurag

Can you explain me the concept of

We need to calculate the number of zeros at the end of X = 50!

and

Hence, number of times X can be completely divided by 100 is (12/2) = 6

[Anurag@Gurome]

We need to calculate the number of zeros at the end of X = 50!

Hence, number of times X can be completely divided by 100 is (12/2) = 6

The question asked for greatest value of k for which 100^k is a factor of X.
This simply means k is the number of times X can be completely divided by 100. For example, if X was equal to 500, then k = 1. If X was equal to 4500000, then k = 2 etc.

Hence, k = The integral part of [(Number of zeroes at the end of X)/2]

Hope it clears your doubt.