Highest power of a number in a Factorial
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The factorial of $$n$$ is represented as $$n!=1*2*3...*(n-1)*n$$
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The highest power of a number $$p$$ in a factorial $$n!$$ if $$p$$ is prime: $$\left[\frac{n}{p}\right]\ +\ \left[\frac{n}{p^2}\right]\ +\ \left[\frac{n}{p^3}\right]\ +\ .....$$
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Let's take the example of the highest power of 5 in 30!
To find the highest power of 5 in 30, find the number of multiples of 5 in 30 + find the number of multiples of 25 in 30 + so on
Hence, the highest power of 5 in 30= 6+1 = 7
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In general, the highest power of p in n is sum = number of multiples of p + number of multiples of $$p^2$$ + number of multiples of $$p^3$$ and so on.
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The highest power of a number c in a factorial n! if c is composite:
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If c is a composite, find its highest prime factor. The power of the composite in n! is the same as the power of its highest prime factor. Hence, the highest power of 21 in 100! is the highest power of 7 in 100!.
Hence, the highest power of 7 in 100! = 14+2=16.
Hence, the highest power of 21 in 100! is 16.
