•To find the highest power of x that divides N!, keep dividing N successively by x and the addition of all the quotients is your answer. (Successive division means dividing the quotient of the earlier division). While this is the process of getting the answer, do understand the concept behind find the highest power as it may be used in other application. The interpretation of the highest power of x that divides a factorial is that if from 1 × 2 × 3 × 4 × 5 × 6 × …… × N, if all the powers of x is segregated, how many will they amount to. E.g. From 19!, if we segregate all powers of 3 we will have something as follows:

1×2×3×4×5×(3×2)×7×8×(3×3)×10×11×(3×4)×13×14×(3×5)×16×17×(3×3×2)×19
i.e. 38 × N, where N will not be a multiple of 3.

The same logic as above can be used for any product and not necessarily a factorial. Thus if the questions is what is the highest power of 3 that can divide the product of squares of all odd numbers from 1 to 20……one should identify that the powers of 3 would appear only in the squares of 3, 9 and 15. Thus the largest power of 3 dividing the given product is 2 + 4 + 2 = 8.

Consider another method
Suppose, we need to find the highest power of 5 that divides 100!
Solution:
100 / 5 = 20
20 / 5 = 4
4 / 5 = 0
Hence the total contribution of the powers of 5 is 24.
Or the number 100! is divisible by 5^24.

•When the number of zeroes at the end of a product of series of numbers is asked, think of the highest power of 2 and 5 in the product.

•If any number is expressed as 10n × m, where m is not a multiple of 10, then n is the number of zeroes at the end of the given number. n is also the highest power of 10 that divides the given number.

•The above property is not just limited to 10. If a number N can be expressed as 7n × m, where m is not a multiple of 7, then in this case also n is the largest power of 7 that can divide the given number.

•The above rules can be used effectively to factorize a factorial. Thus if one needs to find the number of ways in which 15! can be written as a product of 2 numbers……Recollect from the earlier matter that to find the number of ways a number can be written as a product of two natural numbers, one has to find the number of factors of the given number. Also recollect that to find the number of factors of any given number one has to factorize the given number. Factorizing the given number means writing it in the form of 2a × 3b × 5c × 7d × 11e × ……i.e. writing it in powers of prime numbers. Thus we need to find the highest power of prime numbers that can divide the given number. Thus in our case 15! can be written as 211 × 36 × 53 × 71 × 111 × 131. Now one can easily find the number of factors and then half them to get to the answer.