Question

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



a. 14



b. 15



c. 28



d. 29?

Answer 1

16^n=(2^3)^n

44^44=(11*2^2)^44=11^44*2^88

(11^44 *2^88)/ 2^3n

3n=88

n=29

Here 3n=88,n=29. Hence ans will be 29 bcoz 29 is the greatest possible positive integer without leaving remainder.

Answer 2

Given:

$\frac{(44)^{44}}{16^n}$

To find:

Greatest possible positive integer value of n if $16^n$ divides $(44)^{44}$ without leaving a remainder.

Solution:

$16=2\times 2\times 2\times 2=2^4$

$44=11\times 2^2$

$\frac{(11\times 2^2)^{44}}{(2^4)^n}$

$\frac{(11)^{44}\times 2^{2\times 44}}{2^{4n}}$

Using identity

$(a\times b)^n=a^n\times b^n$

$(a^x)^y=a^{xy}$

$\frac{(11)^{44}\times 2^{88}}{2^{4n}}$

If $16^n$ divides $(44)^{44}$ without leaving a remainder

Therefore,

$2^{4n}=2^{88}$

$4n=88$

$n=\frac{88}{4}=22$

Hence, greatest possible positive integer value of n=22