Question

For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained.
a) 252
b) 180
c) 1008
d) 2028
e) 1458
f) 768

Answer 1

Hint:Before solving this question, let us know about Perfect squares and Square roots.
PERFECT NUMBERS: A square number or perfect square is an integer that is obtained after multiplying a number by itself. In other words, it is the product of an integer with itself. For example, 9 is a square number, because it can be written as \[3\times 3\] .
SQUARE ROOT: A square root of a number is a value that when multiplied by itself, gives the number.
Example: \[4\times 4=16\] , so a square root of 16 is 4.
For solving this question, we will do the prime factorization of the given numbers.

Complete step-by-step answer:
252
              \[\begin{align}
  & 2\overline{\left){252}\right.} \\
 & 2\overline{\left){126}\right.} \\
 & 3\overline{\left){63}\right.} \\
 & 3\overline{\left){21}\right.} \\
 & 7\overline{\left){7}\right.} \\
 & \overline{\left){1}\right.} \\
\end{align}\]

              \[252\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }7\]
We can see that 7 does not occur in pairs, so we will multiply 252 by 7 for making it a perfect square.

\[\begin{array}{*{35}{l}}
   252\text{ }\times \text{ }7\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }7\text{ }\times \text{ }7 \\
   1764\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }7\text{ }\times \text{ }7 \\
\end{array}\]
Now, for calculating the square root of 1764, we will take one number from each pair and multiply them.
Square root of 1764 = \[2\text{ }\times \text{ }3\text{ }\times \text{ }7\text{ }=\text{ }6\text{ }\times \text{ }7\] = 42
Smallest number to be multiplied: 7
Square root of the number obtained: 42
180
\[\begin{align}
  & 2\overline{\left){180}\right.} \\
 & 2\overline{\left){90}\right.} \\
 & 3\overline{\left){45}\right.} \\
 & 3\overline{\left){15}\right.} \\
 & 5\overline{\left){5}\right.} \\
 & \overline{\left){1}\right.} \\
\end{align}\]
              \[180\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }5\]
We can see that 5 do not occur in pairs, so we will multiply 180 by 5 to make it a perfect square.
 \[\begin{array}{*{35}{l}}
   180\text{ }\times \text{ }5\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }5\text{ }\times \text{ }5 \\
   900=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }5\text{ }\times \text{ }5 \\
\end{array}\]
Now, for calculating the square root of 1764, we will take one number from each pair and multiply them.
Square root of 900 = \[2\text{ }\times \text{ }3\text{ }\times \text{ }5\text{ }=\text{ }6\text{ }\times \text{ }5\] = 30
Smallest number to be multiplied: 5
Square root of the number obtained: 30

1008
\[\begin{align}
  & 2\overline{\left){1008}\right.} \\
 & 2\overline{\left){504}\right.} \\
 & 2\overline{\left){252}\right.} \\
 & 2\overline{\left){126}\right.} \\
 & 3\overline{\left){63}\right.} \\
 & 3\overline{\left){21}\right.} \\
 & 7\overline{\left){7}\right.} \\
 & \overline{\left){1}\right.} \\
\end{align}\]
1008 = \[2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }7\]
We can see that 7 do not occur in pairs, so we will multiply 1008 by 7 to make it a perfect square.
\[\begin{array}{*{35}{l}}
   1008\text{ }\times \text{ }7\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }7\text{ }\times \text{ }7 \\
   7056\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }7\text{ }\times \text{ }7 \\
\end{array}\]
Now, for calculating the square root of 7056, we will take one number from each pair and multiply them.
Square root of 7056 = \[2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }7\text{ }=\text{ }14\text{ }\times \text{ }6\] = 84
Smallest number to be multiplied: 7
Square root of the number obtained: 84

2028
\[\begin{align}
  & 2\overline{\left){2028}\right.} \\
 & 2\overline{\left){1014}\right.} \\
 & 3\overline{\left){507}\right.} \\
 & 13\overline{\left){169}\right.} \\
 & 13\overline{\left){13}\right.} \\
 & \overline{\left){1}\right.} \\
\end{align}\]
2028 = \[2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }13\text{ }\times \text{ }13\]
We can see that 3 does not occur in pairs, so, we will multiply 2028 by 3 to make it a perfect square.
\[\begin{array}{*{35}{l}}
   2028\text{ }\times \text{ }3\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }13\text{ }\times \text{ }13 \\
   6084\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }13\text{ }\times \text{ }13 \\
\end{array}\]
Now, for calculating the square root of 6084, we will take one number from each pair and multiply them.
Square root of 6084 = \[~2\text{ }\times \text{ }3\text{ }\times \text{ }13\text{ }=\text{ }6\text{ }\times \text{ }13\] = 78
Smallest number to be multiplied: 3
Square root of the number obtained: 78

1458
\[\begin{align}
  & 2\overline{\left){1458}\right.} \\
 & 3\overline{\left){729}\right.} \\
 & 3\overline{\left){243}\right.} \\
 & 3\overline{\left){81}\right.} \\
 & 3\overline{\left){27}\right.} \\
 & 3\overline{\left){9}\right.} \\
 & 3\overline{\left){3}\right.} \\
 & \overline{\left){1}\right.} \\
\end{align}\]
1458 = \[2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3\]
We can see that 2 do not occur in pairs, so, we will multiply 1458 by 2 to make it a perfect square.
\[\begin{array}{*{35}{l}}
   1458\text{ }\times \text{ }2\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3 \\
   2916\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3 \\
\end{array}\]
Now, for calculating the square root of 2916, we will take one number from each pair and multiply them.
Square root of 2916 = \[2\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }\times \text{ }3\text{ }=\text{ }6\text{ }\times \text{ }9\text{ }=\text{ }54\]
Smallest number to be multiplied: 2
Square root of the number obtained: 54

768
\[\begin{align}
  & 2\overline{\left){768}\right.} \\
 & 2\overline{\left){384}\right.} \\
 & 2\overline{\left){192}\right.} \\
 & 2\overline{\left){96}\right.} \\
 & 2\overline{\left){48}\right.} \\
 & 2\overline{\left){24}\right.} \\
 & 3\overline{\left){12}\right.} \\
 & 2\overline{\left){6}\right.} \\
 & 3\overline{\left){3}\right.} \\
 & \overline{\left){1}\right.} \\
\end{align}\]

768 = \[~2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\]
We can see that 3 does not occur in pairs, so we will multiply 768 by 3 for making it a perfect square.
\[\begin{array}{*{35}{l}}
   768\text{ }\times \text{ }3\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3 \\
   2304\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }\times \text{ }3 \\
\end{array}\]
Now, for calculating the square root of 2304, we will take one number from each pair and multiply them.
Square of 2304 = \[2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\text{ }=\text{ }4\text{ }\times \text{ }4\text{ }\times \text{ }3\text{ }=\text{ }16\text{ }\times \text{ }3\] = 48
Smallest number to be multiplied: 3
Square root of the number obtained: 48

Note:One must do all the calculations very carefully for solving this question.
Also, not only in this question, one must be very careful while doing such questions as any mistake in the calculations can make the answer wrong.