A finite number of digits.

Algebra Level 3

How many digits are there in $\overline{(2^{3000})(5^{3000})}$ ?

Clarification: $\overline{(2^{3000})(5^{3000})}$ is such a number that $2^{3000}$ is in the tenths and $5^{3000}$ is in the units.

The answer is 3001.

1 solution

Mr. India
Mar 21, 2019

Let $2^{3000}=10^x$

Then, $x=\log_{10} 2^{3000}$

Or, $x=3000\log_10 2$

Similarly, for $5^{3000}=10^y$ , we have $y=3000\log_{10} 5$

Number of digits in $10^n=n+1$

So, number of digits in given number is

$x+1+y=3000(\log_{10} 2+\log_{10} 5)+1$
( $\log_c a+\log_c b=\log_c {ab}$ )

$=3000\log_{10} 10+1$

$=\boxed{\boxed{3001}}$

Actually, you would be right if there weren't any overlines. Could you find a solution after reading the clarification? The answer is coincidentally the same!

Tin Le - 2 years, 2 months ago

A finite number of digits.

The answer is 3001.

1 solution

0 pending reports