Question

What is the smallest positive integer which you can multiply to 75600 to make it a perfect square?

Answer 1

Answer:

The given number is 1352

But, 1352=2×2×13×13×2=2

2

×13

2

×2

To get a perfect square, the prime numbers should occur even no. of times.

Here, 2 occurs only three times.

Thus, we have to multiply 1352 by 2 to get a perfect square.

Step-by-step explanation:

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Answer 2

Given:

Number 75600

To find:

Smallest positive integer to multiply with 75600 to make it a perfect square.

Solution:

We first find the factors of 75600.

$75600=2*2*2*2*3*3*3*5*5*7$

When a number is multiplied by itself, the product obtained is called perfect square.

In the factors of $75600$, we see two pairs of $2$, one pair of $3$ and one pair of $5$. The remaining factors are only present in single, i.e., single factors $3$ and $7$ are remaining. If we multiply another $3$ and $7$ with them, we get one pair of $3$ and one pair of $7$ as (extra) factors. Multiply $3$ and $7$ with $75600$ too to obtain a product of $1587600$ which is a perfect square of $1260$.

Hence, $3*7=21$ is the smallest positive integer that should be multiplied with $75600$ in order to make it a perfect square.

$\sqrt{75600*7*3} =\sqrt{2*2*2*2*3*3*3*3*5*5*7*7}$

$\sqrt{1587600} =2*2*3*3*5*7$

$\sqrt{1587600} =1260$

The smallest positive integer that should be multiplied with $75600$ in order to make it a perfect square is $21$.