Question

is 9 the smallest number which must be multiplied with 10125 to get the product as a perfect square?

Answer 1

Answer:

No,9 is not the smallest number.please mark my answer as brainliest

Answer 2

To know it first we have to do prime factorisation of 10125, making it as product of prime numbers only.

$\begin{tabular}{r|l}\sf{5}&\sf{10125}\\\cline{2-}\sf{5}&\sf{2025}\\\cline{2-}\sf{5}&\sf{405}\\\cline{2-}\sf{3}&\sf{81}\\\cline{2-}\sf{3}&\sf{27}\\\cline{2-}\sf{3}&\sf{9}\\\cline{2-}&\sf{3}\\\cline{2-}\end{tabular}$

Therefore,

$\longrightarrow\sf{10125=5\times5\times5\times3\times3\times3\times3\times3}$

$\longrightarrow\sf{10125=5^3\times3^4}$

We see 10125 is the product of third power of 5 and 4th power of 3.

To get a number as a perfect square, it should be expressed as the product of even powers of prime numbers. If a number has an odd power of a prime factor, then we have to multiply it with the same prime factor to make it a perfect square.

Here third power of 5 is as the factor of 10125. So if we multiply 10125 by 5, it will have 4th power of 5 as factor and thus 10125 multiplied with 5 will be a perfect square.

Hence the least number which must be multiplied with 10125 to get the product a perfect square is 5, but not 9.