Question

By what number should each of the following numbers be multiplied to get a perfect square in each case? Also, find the number whose square in the new number is 2880.

Answer 1

Hint: Here, we have to first find the factors of the given number and the pair the factors to get the perfect square of these factors, if you get one factor leftover, multiply the given number with the leftover factor to get a perfect square for all the factors. In the end, take each number from the perfect square of the factors and multiply them to get the required number (perfect square). For example, 25 x 25 = 625, 8 x 8 = 64, 12 x 12 = 144, these are examples of perfect squares

Complete step-by-step answer:
We have been given the number 2880, first let us find out its factors
$\begin{align}
  & 2\left| \!{\underline {\,
  2880 \,}} \right. \\
 & 2\left| \!{\underline {\,
  1440 \,}} \right. \\
 & 2\left| \!{\underline {\,
  720 \,}} \right. \\
 & 2\left| \!{\underline {\,
  360 \,}} \right. \\
 & 2\left| \!{\underline {\,
  180 \,}} \right. \\
 & 2\left| \!{\underline {\,
  90 \,}} \right. \\
 & 3\left| \!{\underline {\,
  45 \,}} \right. \\
 & 3\left| \!{\underline {\,
  15 \,}} \right. \\
 & 5\left| \!{\underline {\,
  5 \,}} \right. \\
 & \,\,\,1 \\
\end{align}$
From the above, we have the factors for 2880,
2880 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5
Now, let us pair them in groups to find how many perfect squares we have in the factors of 2880
2880 = (2 x 2) x (2 x 2) x (2 x 2) x (3 x 3) x 5
Here, after pairing the prime factors we see that we have the factor 5 without a pair, hence this cannot be a perfect square.
In order to get a perfect square, we need to multiply the number 2880 by 5 on both the sides, and the factors we will get are
2880 x 5 = (2 x 2) x (2 x 2) x (2 x 2) x (3 x 3) x (5 x 5)
We can also write the above factors in this way,
14400 = (2 x 2 x 2 x 3 x 5) x (2 x 2 x 2 x 3 x 5)
We took one factor from each pair to get us a perfect square.
Now, 2 x 2 x 2 x 3 x 5 = 120.
Hence, to make the number 2880 a perfect square, it is to be multiplied with 5, and the perfect square we will get is 120. Therefore, the number 2880 changes to 14400.

Note: In this question, knowing how to get the factors is important. For example, the number 2 is a factor for the numbers whose unit place has even numbers (0, 2, 4, 6, 8). The number 3 is a factor for the numbers when the sum of the digits in the number is a multiple of 3.