Question

The smallest number by which 12348 must be divided to obtain a perfect square is
A.3
B.4
C.5
D.7

Answer 1

Hint: Here, first, we will factorize the given number using prime factorization method. We will then find the smallest number which must be divided to obtain a perfect square. A number which when multiplied by itself two times, then the obtained number must be a perfect square.

Complete step-by-step answer:
We have to find the smallest number by which 12348 must be divided to obtain a perfect square.
First, we will now factorize 12348 using the prime factorization. Prime factorization is a method of factoring a number such that the given number can be broken down in terms of prime factors.
\[\begin{array}{l}{{2}}\left| \!{\underline {\,
  {{{12348}}} \,}} \right. \\{{2}}\left| \!{\underline {\,
  {{{6174}}} \,}} \right. \\{{3}}\left| \!{\underline {\,
  {{{3087}}} \,}} \right. \\{{3}}\left| \!{\underline {\,
  {{{1029}}} \,}} \right. \\{{7}}\left| \!{\underline {\,
  {{{343}}} \,}} \right. \\{{7}}\left| \!{\underline {\,
  {{{49}}} \,}} \right. \\{{7}}\left| \!{\underline {\,
  {{7}} \,}} \right. \end{array}\]
So, we can write 12348 as:
\[12348 = 2 \times 2 \times 3 \times 3 \times 7 \times 7 \times 7\]
Since all the factors of 12348 are in pairs except 7.
So, the number should be divided by 7.
Therefore, the smallest number by which 12348 must be divided to obtain a perfect square is 7.

Note: We know that to find the square root of a number, all the factors should be in pairs. We have to group the factors and the factor which does not have a pair has to be multiplied to obtain a perfect square number. Thus, we obtain the smallest number to be multiplied.
Similarly, we have to group the factor for the numbers, the factor which does not have a pair, that factor has to be divided to obtain a perfect square number. Thus we obtain the smallest number to be divided.