Question

Find the largest perfect square factor in $ 729 $

Answer 1

Hint: For the largest perfect square factor, start with the factorization of the given number. Factorization is the process of breaking down the given composite numbers into the product of the smaller integers. Furthermore, if these integers are breakdown to prime numbers then it is called prime factorization. The perfect square is the number which can be expressed as the product of the two equal integers. For example: $ 9 $ , it can be expressed as the product of equal integers. $ 9 = 3 \times 3 $

Complete step-by-step answer:
Find factors of $ 729 $
Start dividing the given number by $ 3 $ as the given number is the odd number.
 $ \therefore 729 = 243 \times 3 $
Again, dividing the number,
 $
  243\;{\text{by 3}} \\
  \therefore {\text{729 = 81}} \times {\text{3}} \times {\text{3}} \\
  \therefore {\text{729}} = 9 \times 9 \times 3 \times 3 \\
  $ [Breaking down the numbers to least factorization]
Now, according to the properties of the perfect square, making the pairs of the numbers
 $
   \Rightarrow 729 = \underline {9 \times 3} \times \underline {9 \times 3} \\
   \Rightarrow 729 = 27 \times 27 \\
   \Rightarrow 729 = {27^2} \\
  $
Hence, the required answer - the largest perfect square of $ 729\;{\text{is 27}} $ .

Note: Perfect square number is the square of an integer, simply it is the product of the same integer with itself. For example - $ 25{\text{ = 5 }} \times {\text{ 5, 25 = }}{{\text{5}}^2} $ , generally it is denoted by n to the power two i.e. $ {n^2} $ . The factorization can be done by long division method or by the factor tree method. A factor tree number is the special diagram where the number is breakdown and ends with all the prime factors of the original number until more factorization is not possible. Always read the question twice as there is the major difference between squares and square roots.