Question

Let K=$20^{20}$.Suppose that $\frac{20^{k} }{k^{20} } =20^{n}$.find the largest power of 20 that divides n?

Answer 1

Answer:

  20^2 = 400, the 2nd power of 20

Step-by-step explanation:

Given that k=20^20 and 20^k/k^20 = 20^n, you want the largest power of 20 that divides n.

Logarithms

Taking the base-20 logarithm of both equations, we have ...

  $\log_{20}{k}=\log_{20}{20^{20}}\ \Longrightarrow\ \log_{20}{k}=20\\\\\log_{20}{\dfrac{20^k}{k^{20}}}=\log_{20}{20^n}\ \Longrightarrow\ k-20\log_{20}{k}=n$

Substituting for k and log(k), we get ...

  $20^{20} -20\cdot20=n\\\\20^2(20^{18}-1)=n$

This shows us the largest power of 20 that is a factor of n is 20².