Question

State whether \[1335\] is a perfect square or not.

Answer 1

Hint:We solve for the square root of the given number using the division method and check if it is a perfect square.
* A number is a perfect square if it’s under root or square root is a whole number.

Complete step-by-step answer:
First we pair the digits in the number in pair of two each and the remaining one(if any) starting from the right side.
\[1335 = \overline {13} \overline {35} \]
Then we take the highest number whose square will be less than or equal to the first pair i.e. \[13\]
So, we know, \[1 \times 1 = 1,2 \times 2 = 4,3 \times 3 = 9,4 \times 4 = 16\]
We choose \[3 \times 3 = 9\] because \[9 < 13\]
Now we divide the number by taking this number as divisor and taking the same number as quotient.
\[
  3\mathop{\left){\vphantom{1{\overline {13} \overline {35} }}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{\overline {13} \overline {35} }}}}
\limits^{\displaystyle \,\,\, 3} \\
   - 9 \\
  \overline { = 535} \\
 \]
Now the remainder becomes the next dividend and the new divisor is twice the old divisor followed by a digit which makes a number such that square of that number will be less than or equal to the new dividend.
So, we have new dividend as \[535\] and we can have divisor as \[2 \times 3\underline {} = 6\underline {} \]where blank is filled by a digit.
Now we try to find a number in the lane of sixties whose square is less than or equal to our new dividend.
\[
  61 \times 1 = 61 \\
  62 \times 2 = 124 \\
  63 \times 3 = 189 \\
  64 \times 4 = 256 \\
  65 \times 5 = 325 \\
  66 \times 6 = 396 \\
  67 \times 7 = 469 \\
  68 \times 8 = 544 \\
 \]
We can clearly see that \[67 \times 7 = 469\] suits our requirement because \[469 < 535\]
Now we divide the dividend by the number \[67\]and the quotient \[7\]comes beside the earlier quotient.
\[
  67\mathop{\left){\vphantom{1{535}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{{535}}}}
\limits^{\displaystyle \,\,\, {37}} \\
   - 469 \\
  \overline { = 66} \\
 \]
Now we have the remainder \[66\]which is our new dividend and the new divisor is \[2 \times 37 = 74\] and \[74 > 66\] therefore \[74\underline {} \] along with the blank to be filled can never divide the number \[66\]
We cannot write the number \[1335\] in terms of a whole number after taking the square root.
So, the number \[1335\] is not a perfect square.

Note:Alternative method:
We can start calculating squares of numbers from \[1\] and check in the list but looking at the number we can use hit and trial method to find near about squares
Say we know \[30 \times 30 = 900,40 \times 40 = 1600\] So we check numbers in between \[30,40\]
\[35 \times 35 = 1225,36 \times 36 = 1296,37 \times 37 = 1369\]
We see that the number lies between \[1296,1369\] and both are perfect squares of consecutive numbers, which means there is no whole number whose square is \[1335\].