recategorized by
11,482 views
29 votes
29 votes

$X$ is a $30$ digit number starting with the digit $4$ followed by the digit $7$. Then the number $X^3$ will have

  1. $90$ digits
  2. $91$ digits
  3. $92$ digits
  4. $93$ digits
recategorized by

6 Answers

Best answer
101 votes
101 votes
$X = 4777\dots $ ( $7$ $29$ times)

It can be written as $X = 4.7777\dots *10^{29}$

$X^3 = (4.777\dots*10^{29})^3 = (4.777\dots)^3* 10^{87}$

Now, even if we round up $4.777\dots$ to $5$, we could represent $5^3 = 125$ in $3$ digits. So, We can say $(4.77\dots)^3$ also has $3$ digits before decimal point.

So, $X^3 $ requires $3 + 87 = 90$ digits.

Correct Answer: $A$
edited by
11 votes
11 votes

Here's a simple approach purely based on observation:

Note that $47^{3}$ is $103823$ (6 digits)

Start with a smaller number, say $4700$ : It's cube will be $103823$ followed by $000000$.
How about $4799$? It's cube will be $110522$ followed by $894400$. Did number of digits increase? No.

Trying with more number of digits after 47, we can see that the first part $47^{3}$ will always give 6 digits and the remaining number of digits after it are getting tripled in number, eg for two digits after $47$ we got six digits in the cube of $4700$ as shown above.

So extending it for $47$ followed by $28$ more digits, we will get $6$ digits for $47^{3}$ and $3*28$ more digits for the number after $47$.

We get total : $6 + 84 = 90$ digits.

Option A. 

edited by
4 votes
4 votes

Here’s the exact and quick approach:

Simply we know that in scientific notation say

(a)x=4.777777…. *$10^{29}$

(b)10x= 47.77777…. *$10^{29}$

Substracting b-a:

9x=43 *$10^{29}$

x= $\frac{43*10^{29}}{9}$

$x^{3}$= 109.063*$10^{87}$

So it has 3+87=90 digits

Answer:

Related questions

10 votes
10 votes
6 answers
1
16 votes
16 votes
4 answers
2
Arjun asked Feb 14, 2017
5,002 views
Find the smallest number $y$ such that $y \times 162$ is a perfect cube.$24$$27$$32$$36$
11 votes
11 votes
2 answers
3
makhdoom ghaya asked Jan 25, 2017
4,201 views
The numeral in the units position of $211^{870}+146^{127} \times 3^{424}$ is ________.