Question

What smallest number must be added to 269 to make it a perfect square?
A.\[31\]
B.\[16\]
C.\[7\]
D.\[20\]

Answer 1

Hint: Here we will discuss the concept of perfect squares in detail. Also, we will discuss the method of converting a given number into a perfect square. Later we will use the concept of perfect square to get the least number that must be added to make 269 a perfect square.
Perfect square: If we multiply a number by itself then we obtain a number which is a perfect square. For example, let’s find the perfect square of \[7\], its perfect square is given as, \[7 \times 7 = 49\]. Similarly, there are many perfect squares of respective numbers such as \[125\] is a perfect square formed from number \[5\] ,\[36\] is formed from number \[6\]etc.

Complete step-by-step solution:
The most important property of perfect squares is that they are always positive, moreover perfect square forms of negative numbers are also positive. If we find the square root of a perfect then we obtain the number from which it was formed. For example, consider the perfect square \[64\], its square root is given as, \[\sqrt {64} = 8\].
Now, we are asked to determine the smallest number that must be added to \[269\] to make it a perfect square.
For doing so we will find the nearest perfect square which is just greater than the number \[269\].
We know that the nearest perfect square is \[289\] which is formed from number \[7\]. Since, we are asked to determine the number that has to be added therefore we will subtract the number \[269\] from \[289\], we obtain \[289 - 269 = 20\].

Hence, option D is the correct option.

Note: In such types of problems we must find the nearest perfect square to the given number. If we are asked to determine the number that has to be added then we should subtract the given number from the perfect square and if we are asked to determine the number which has to be subtracted then we must subtract the perfect square from the given number.