Question

What is the greatest possible positive integer n if 16^n divides (44)^44 without leaving a remainder?

Answer 1

Answer: n=22

Step-by-step explanation:

The law of exponents are given by:-

$a^n\times a^m=a^{n+m}\\\\\frac{a^m}{a^n}=a^{m-n}\\\\(a^m)^n=a^{mn}\\\\a^n\timesb^n=(ab)^n$

By using above law of exponents we have

$(44)^{44}\\=(4\times11)^{44}\\=4^{44}\times11^{44}\\=4^{2\times22}\times11^{44}\\=(4^2)^{22}\times11^{44}\\=16^{22}\times11^{44}$

Divide $(44)^{44}$ by $16^n$ , where n be any positive integer.

$\frac{(44)^{44}}{16^n}=\frac{16^{22}\times11^{44}}{16^n}$

To divide it without leaving a remainder, the value of n should be 22 such that

$\frac{(44)^{44}}{16^{22}}=\frac{16^{22}\times11^{44}}{16^{22}}=11^{44}$