Question

Using prime factorisation, find which of the following are perfect squares.
(a) \[484\]
(b) \[11250\]
(c) \[841\]
(d) \[729\]

Answer 1

Hint:If a natural number \[m\] can be expressed as \[{{n}^{2}}\] where \[n\] is also a natural number then \[m\] is a square number. The numbers \[1,4,9,16,25,\ldots \ldots \] are perfect square numbers. These numbers are called perfect squares. Prime factorisation is the method of expressing the given number as the product of prime factors. \[4\] is the perfect square of \[2\] and \[9\] is the perfect square of \[3\]. Prime factorisation of \[4\], \[4=2\times 2\] and \[9\], \[9=3\times 3\].

Complete step by step answer:
(a) Prime factorisation of \[484\]
Express \[484\] as the product of prime factors
\[484=2\times 2\times 11\times 11\]
Keep these factors in pairs. Take the product of the prime factors by only taking one out of every pair of the same primes. This product gives the square root of the number.
\[\sqrt{484}=2\times 11\]
\[\Rightarrow \sqrt{484}=22\]
If any prime factor does not have any pair then the number is not a perfect square and if all the factors are in pair then the number is a perfect square. Hence \[484\] is a perfect square.

(b) Prime factorisation of \[11250\]
Express\[11250\] as the product of prime factors
\[11250=2\times 3\times 3\times 5\times 5\times 5\times 5\]
Keep these factors in pairs. Take the product of the prime factors by only taking one out of every pair of the same primes. This product gives the square root of the number. Here the prime factor \[2\] is not in pair. If any prime factor does not have any pair then the number is not a perfect square and if all the factors are in pair then the number is a perfect square. Hence \[11250\] is not a perfect square.

(c) Prime factorisation of\[841\]
Express \[841\] as the product of prime factors
\[841=29\times 29\]
Keep these factors in pairs. Take the product of the prime factors by only taking one out of every pair of the same primes. This product gives the square root of the number.
\[\sqrt{841}=29\]
If any prime factor does not have any pair then the number is not a perfect square and if all the factors are in pair then the number is a perfect square. Hence \[841\] is a perfect square.

(d) Prime factorisation of \[729\]
Express \[729\] as the product of prime factors
\[729=3\times 3\times 3\times 3\times 3\times 3\]
Keep these factors in pairs. Take the product of the prime factors by only taking one out of every pair of the same primes. This product gives the square root of the number.
\[\sqrt{729}=3\times 3\times 3\]
\[\Rightarrow \sqrt{729}=27\]
If any prime factor does not have any pair then the number is not a perfect square and if all the factors are in pair then the number is a perfect square. Hence \[729\] is a perfect square.

Note: A perfect square number cannot have an odd number of zeroes at its end. It cannot end with \[2,3,7or8\]. Squares of even numbers are always even and squares of odd numbers are always odd. The square root of a number is that number which when multiplied by itself will give that number.