Problem Solving

satyavegi wrote: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.

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.