Question

(I) Find all square numbers between 60 and 120.
(II) What smallest number should be added to 250 to make it a square number?

Answer 1

Hint: In this type of question we have to use the concept of a perfect square number. We know that any number multiplied by itself creates a perfect square.
In the first part of the question we consider all the numbers ending with the digits 1, 4, 5, 6, 9 or 0 as we know that a number is a perfect square if at unit place 1, 4, 5, 6, 9 or 0 is present. Then we check whether they are perfect squares or not. To check whether the given number is a perfect square or not we will use the definition of perfect square.
Now in the second part of the question we have given one number and we have to convert it into a perfect square by adding the smallest possible number. We first find out the nearest perfect square and then we perform subtraction of the perfect square and given number to obtain the required result.

Complete step by step solution:
(I) Now, we have to find all the square numbers between 60 and 120.
As we know, a number is a perfect square if it ends with the digit 0, 1, 4, 5, 6 or 9. So let us consider the numbers between 60 and 120 which ends with the digit 0, 1, 4, 5, 6 or 9. Hence the corresponding numbers are as follows:
\[\begin{align}
  & 61,64,65,66,69,70,71,74,75,76,79,80,81,84,85,86,89,90,91,94,95,96, \\
 & 99,100,101,104,105,106,109,110,111,114,115,116,119 \\
\end{align}\]
Now, from the above list we can observe that, \[61,71,79,89,101,109\] are the prime numbers and hence cannot be square numbers.
Now, for the remaining numbers we will consider the factorisation,
\[\begin{align}
  & \Rightarrow 64=8\times 8,65=13\times 5,66=2\times 3\times 11,69=3\times 23,70=2\times 5\times 7, \\
 & 74=2\times 37,75=3\times 5\times 5,76=2\times 2\times 19,80=2\times 2\times 2\times 2\times 5,81=9\times 9, \\
 & 84=2\times 2\times 3\times 7,85=5\times 17,86=2\times 43,90=2\times 3\times 3\times 5,91=7\times 13, \\
 & 94=2\times 47,95=5\times 19,96=2\times 2\times 2\times 2\times 2\times 3,99=3\times 3\times 11, \\
 & 100=2\times 2\times 5\times 5=10\times 10,104=2\times 2\times 2\times 13,105=3\times 5\times 7,106=2\times 53, \\
 & 110=2\times 5\times 11,111=3\times 37,114=2\times 3\times 19,115=5\times 23, \\
 & 116=2\times 2\times 29,119=7\times 17 \\
\end{align}\]
Now, as we know that, any number multiplied by itself creates a perfect square. We will look at the numbers who have their highest factors repeated twice.
From the above list we can observe that there are three such numbers viz. 64, 81 and 100.
Hence, 64, 81 and 100 are square numbers between 60 and 120.
(II) Now, we have to find the smallest possible number such that if we add it to \[250\] we will get a perfect square.
Let us find the nearest integer to \[250\] such that it is a perfect square and is just greater than the number \[250\].
We know that the nearest perfect square is \[256\] which is formed from the number \[16\].
Since, we are asked to determine the number that has to be added to \[250\] therefore we will subtract 250 from \[256\],
\[\Rightarrow 256-250=6\]
Hence, we have to add \[6\] to \[250\]to obtain a perfect square.

Note: In the first part of the question students have to remember that the perfect squares ends with the digits 0, 1, 4, 5 ,6 or 9 otherwise they have to remember the squares of the numbers from 1 to 10. Also students have to be well familiar with the concept of prime numbers. One of the students may solve as follows:
Let us list down squares of all numbers from 1 to 11
\[\begin{align}
  & \Rightarrow {{1}^{2}}=1,{{2}^{2}}=4,{{3}^{2}}=9,{{4}^{2}}=16,{{5}^{2}}=25,{{6}^{2}}=36 \\
 & {{7}^{2}}=49,{{8}^{2}}=64,{{9}^{2}}=81,{{10}^{2}}=100,{{11}^{2}}=121 \\
\end{align}\]
From the above list we can observe that in between 60 to 120 only three squares lie 64, 81 and 100.
In the second part of the question students have to find the nearest perfect square to the given number. Also, students have to take care that as we are asked to find the number that has to be added to the given number then we should subtract the given number from the perfect square and if we are asked to find the number that has to be subtracted from the given number then we should add the given number to the perfect square.