Question

The highest power of 15 in 130!

Answer 1

15 = 5 × 3

So we first find the highest power of 5 in 130!. Then highest power of 3 in 130! is found. The smallest among them will be the answer.

We have the formula for finding the highest power of a prime number in n!.

$\displaystyle E_p(n!)=\sum_{i=1}^k\left[\dfrac {n}{p^i}\right]\ \text{where}\ \ k\leq\log_pn\ \textless\ k+1$

and [a] is the greatest integer below a.

So the highest power of 5 in 130! is,

$\left [\dfrac {130}{5}\right]+\left [\dfrac {130}{5^2}\right]+\left [\dfrac {130}{5^3}\right]\\\\\\=26+5+1=32$

And the highest power of 3 in 130! is,

$\left [\dfrac {130}{3}\right]+\left [\dfrac {130}{3^2}\right]+\left [\dfrac {130}{3^3}\right]+\left [\dfrac {130}{3^4}\right]\\\\\\=43+14+4+1=62$

Here 32 is the least, hence 32 is the answer.

So the highest power of 15 in 130! is 15^32.