For what value of N will 270N be a perfect sq | vedclass

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(NONE) First,find the prime factorization of $270$: $270 = 27 	imes 10 = 3^3 	imes 2 	imes 5 = 2^1 	imes 3^3 	imes 5^1$.
To make the number a pe

Vedclass · Accepted Answer

$30$

Vedclass · Answer

(NONE) First,find the prime factorization of $270$: $270 = 27 	imes 10 = 3^3 	imes 2 	imes 5 = 2^1 	imes 3^3 	imes 5^1$.
To make the number a perfect square,the exponent of each prime factor must be an even number.
Therefore,$N$ must contain at least $2^1 	imes 3^1 	imes 5^1 = 30$ to make the exponents even $(2^2 	imes 3^4 	imes 5^2)$.
If $N = 30$,then $270 	imes 30 = 8100 = 90^2$,which is a $4$-digit number.
Checking the given options:
If $N = 1$,$270 	imes 1 = 270$ (not a perfect square).
If $N = 6$,$270 	imes 6 = 1620$ (not a perfect square).
If $N = 4$,$270 	imes 4 = 1080$ (not a perfect square).
If $N = 9$,$270 	imes 9 = 2430$ (not a perfect square).
Since $N=30$ is the smallest value to make it a perfect square and it is not in the options,there is an error in the provided options. However,based on the logic of perfect squares,$N$ must be of the form $30 	imes k^2$.

For what value of $N$ will $270N$ be a perfect square,where $270N$ is a $4$-digit number?

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