TAL010 wrote:
Finding the powers of a prime number p, in the n!
The formula is:
Example:
What is the power of 2 in 25!?
^^ Taken from the GMAT Club book...what is the logic behind this question? What are they really asking?
It means calculating number of instances of P in n!
Consider the simple example ---> what is the power of 3 in 10!
We can find four instances of three in 10! -----> 1 * 2 *
3 * 4 * 5 * (2*
3) * 7 * 8 * (
3*
3) * 10
You can see above we can get four 3s in the expression.
Calculating the number of instances in this way could be tedious in the long expressions. but there is a simple formula to calculate the powers of a particular prime.
the powers of Prime P in n! can be given by \(\frac{n}{p} + \frac{n}{p^2} + \frac{n}{p^3} + .................\) till the denominator equal to or less than the numerator.
what is the power of 3 in 10! ------> \(\frac{10}{3} + \frac{10}{3^2} = 3 + 1 = 4\)
Analyze how the process works........
We first divided 10 by 1st power of 3 i.e. by 3^1 in order to get all red 3s
Later we divided 10 by 2nd power of 3 i.e. by 3^2 in order to get the leftover 3 (blue)
we can continue in this way by increasing power of
P as long as it does not greater than
nBack to the original question..............
What is the power of 2 in 25!? ---------> 25/2 + 25/4 + 25/8 + 25/16 = 12 + 6 + 3 + 1 = 22
Hope that helps!
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