Question

Find the square of 527 by using the diagonal method.

Answer 1

Hint: To find square of number we will use diagonal method. We will form a square and divide it into sub-square based on the number of digits in the given number. Then we will write all digits separately in front of each row and each column. We will multiply each digit on the left to each digit on the top. Adding digits diagonally starting from the lowest diagonal we will get the required answer.

Complete step-by-step answer:
First, we will make a square and divide it into 9 parts as we have 3 digits. We will multiply the two digits on the left of the row and top most of the column. We will write the number such that the digit at 10th place is above the diagonal and digit at unit place below the diagonal of the sub-square between the two digits.

	\[5\]	\[2\]	\[7\]
\[5\]	\[2\] \[5\]	\[1\] \[0\]	\[3\] \[5\]
\[2\]	\[1\] \[0\]	\[0\] \[4\]	\[1\] \[4\]
\[7\]	\[3\] \[5\]	\[1\] \[4\]	\[4\] \[9\]

Now we will get the square of the number from backward by each diagonal of the square. Starting from the lower most diagonal, we will write the digit. If there is more than one digit we will add them and then right the unit place digit first and add the 10th place digit to the next diagonal as follows:
Digit at the lower most diagonal is 9 so last digit of square is 9
Next the 2nd most diagonal has three digits and on adding them, we get
\[4 + 4 + 4 = 12\]
So the next digit after 9 is 2 and 1. We will add 2 and 1 to the next diagonal which have 5 digits. Therefore, we get
\[5 + 1 + 4 + 1 + 5 + 1 = 17\]
So the next digit after 9 and 2 i.e. 7 and 1 will be added to the next diagonal which has 5 digits.
\[3 + 0 + 0 + 0 + 3 + 1 = 7\]
So the next digit after 9, 2 and 7 is 7.
Next diagonal have three digits and on adding them, we get
\[1 + 5 + 1 = 7\]
So the next digit after 9,2, 7, 7 will be 7.
Next diagonal has only one digit. The number is 2.
So, we get square of 527 as
\[{\left( {527} \right)^2} = 277729\]

Note: Here, we need to write the unit digit below the diagonal and 10th digit above the diagonal. This method is a lengthier method to find squares. Square of the number can also be obtained by multiplying the number by itself. So we can directly multiply 527 to 527 to get the required answer.
\[\begin{array}{l}\begin{array}{*{20}{l}}{527}\\{ \times 527}\end{array}\\\overline {\begin{array}{*{20}{r}}{3689}\\{1054 \times }\\\begin{array}{r}\underline { + 2635 \times \times } \\277729\end{array}\end{array}} \end{array}\]
So, the answer is 277729.