Question

Find the highest power of 16 in 100!

Answer 1

Concept:

Power is defined by a bse number and sn exponent. The base number represents the number that is being multiplied to get the power.

Given:

$16$ in $100!$

Find:

The highest power of $16$ in $100!$

Solution:

According to the problem,

$16=2*8$

This can be simplied more into,

$16=2*2*4$

Using floor function, the highest power of $2$ in $100!$ is

$\frac{100}{2}+\frac{100}{2^2} +\frac{100}{2^3} +\frac{100}{2^4} +\frac{100}{2^5} +\frac{100}{2^6} +\frac{100}{2^7}+...........$

$50+25+12+6+3+1+0=97$

Using floor function, the highest power of $4$ in $100!$ is

$\frac{100}{4}+\frac{100}{4^2} +\frac{100}{4^3} +...........$

$25+6+1=32$

Since $97$ is higher exponent thus,

$2^{32}*2^{32}*4^{32}$

$=16^{32}$ is the highest power

Hence, the highest power of $16$ in $100!$ is $16^{32}$