How many factors of the number 2^3 x 3 x 5^2 are perfect squares?

Solution

• Per our conceptual knowledge, a number is a perfect square only when the exponents of the prime factors in the number are even.

o For example: \(7^2\)is a perfect square but \(5^3\) is not, since the exponent of \(7\)is even but the exponent of \(5\)is not.
• To find the factors which are perfect squares we need to first of focus on each prime number and its powers and try to find out, what are the possible perfect squares that we can create using those prime numbers.
• So, for making perfect squares, we need even powers of the prime factors and this can be written in the form: \(2^a * 3^b * 5^c\) [where a, b and c can be 0,2,4,8…]

o The factors of\(2^3\) are: \(1, 2, 4, 8\) out of which only \(1(2^0)\) and \(4(2^2)\) are perfect squares.
o The factors of \(3\) are: \(1\) and \(3\), out of which only \(1(3^0)\) is a perfect square.
o The factors of \(5^2\)are: \(1, 5, 25,\) out of which only \(1(5^0)\) and \(25(5^2)\) are perfect squares.
• Thus, the possible unique perfect square factors using only these prime numbers are \(1\), \(4\) and \(25\).
But is that all?
Look at the following factor:
• Is \(2^2\) *\(5^2\) a perfect square?

o Yes, it is.
So, the criterion of perfect squares also holds true when we multiply 2 or more such perfect squares together.

o Hence, 100 will also be included in our list.