Question

The least number by which 176 be multiplied to make the result a perfect square?
(A) $11$
(B) $8$
(C) $2$
(D) $3$

Answer 1

Hint: First we must factorise the given number. Next we will make pairs of the resultant factors. After grouping them into pairs, we will see for the number which is not grouped together. Then if we multiply the given number with that number which is not grouped we get the perfect square number.

Complete step-by-step answer:
First we will be dividing the given digit which is 176 into the minimum fraction.
$ \Rightarrow 176 = 2 \times 2 \times 2 \times 2 \times 11$
Since this is how we divided the 176 digit in another form so we can see that there are four 2 multiple digits but there is only one 11 digit so we have to make it two so that we can get the right answer.
Here we got two pairs of 2 and one non grouped 11. So if we multiply the number by 11 then the resultant number will be a perfect square.
$ \Rightarrow 176 = (2 \times 2) \times (2 \times 2) \times 11$
$ \Rightarrow 176 = {2^2} \times {2^2} \times 11$
We have to make this one 11 digit into two and by then we can get the answer which is 11.
To verify the given answer we multiply the given number by the answer we get
$ \Rightarrow 176 \times 11 = 1936$
So the factors of $1936$ is
$ \Rightarrow 2 \times 2 \times 2 \times 2 \times 11 \times 11$
Now by pairing the numbers we get
$ \Rightarrow {2^2} \times {2^2} \times {11^2}$
After solving we get
$ \Rightarrow {(44)^2}$ which is a perfect square.

Hence the correct option is A.

Note: Perfect Square: It is a number made by squaring a whole number. In other words we can say that a perfect square is a number when the given number is multiplied by itself.