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Find the smallest number which should be multiplied to 10976 to get a perfect cube?

seo-qna
Last updated date: 21st Feb 2024
Total views: 337.5k
Views today: 9.37k
IVSAT 2024
Answer
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Hint: To find the smallest number which should be multiplied to 10976 to get a perfect cube, we will first of all find the prime factorization of 10976 and then we will see in the factorization that on multiplication of which number will make this number as perfect cube. For e.g., if prime factorization is like ${{5}^{2}}$ then the smallest number which on multiplication with this number is 5 then the power of 5 becomes 3 and 3 is divisible by 3 and we can get the perfect cube.

Complete step-by-step solution:
The number given above which we have to make perfect cube is as follows:
10976
Now, we are going to write the prime factorization of the above number which is equal to:
$=2\times 2\times 2\times 2\times 2\times 7\times 7\times 7$
Now, we know that if base is same and written with multiplication sign then powers will add up so adding the powers of 2 and the powers of 7 we get,
$\begin{align}
  & ={{2}^{1+1+1+1+1}}\times {{7}^{1+1+1}} \\
 & ={{2}^{5}}\times {{7}^{3}} \\
\end{align}$
As you can see the powers of 2 and 7 then power of 7 is divisible by 3 but power of 2 is not divisible by 3. Power of 2 is 5 and we can make this power divisible by 3 if we add 1 to the power of 2. So, adding 1 in the above power of 2 we get,
$\Rightarrow {{2}^{5+1}}\times {{7}^{3}}$
We know the property when base is same and written in the multiplied form then we can add the powers so vice versa is also true if powers are written in the addition form we can convert them into multiplication so the power in 2 which is 1 and added to 5 we can eliminate this addition of 1 by multiplying ${{2}^{5}}$ by 2 and we get,
$\Rightarrow {{2}^{5}}\times 2\times {{7}^{3}}$
From the above, we have found on multiplying 2 with 10796 we get the perfect cube.
Hence, the smallest number which on multiplication with 10796 will make this number perfect cube is 2.

Note: You can check the smallest number which you have got above is correct or not by multiplying the number 10976 by 2 and then do the cube root of that multiplied number.
Multiplying 10976 by 2 we get,
$\begin{align}
  & \Rightarrow 10976\times 2 \\
 & =21952 \\
\end{align}$
Now, taking cube root of the above number we get,
$\begin{align}
  & \Rightarrow {{\left( 21592 \right)}^{\dfrac{1}{3}}} \\
 & =28 \\
\end{align}$
Hence, we have multiplied the correct smallest number.