Question

Study the pattern and answer the following:

$$\begin{gathered} 15\times 15=\mathbf{2}25 \\ 25\times 25=\mathbf{6}25 \\ 35\times 35=\mathbf{12}25 \\ 45\times 45=\mathbf{20}25 \end{gathered}$$

$$\begin{gathered} 55\times 55=....25 \\ 65\times 65=....25 \\ 75\times 75=....25 \\ 85\times 85=....25 \\ 95\times 95=....25 \\ 105\times 105=....25 \end{gathered}$$

Answer 1

Finding the missing numbers:

By observing the pattern, we get

All the numbers are ending with $25$, other than $25$ the digits are changing.

Lets observe $25\times 25=\mathbf{6}25$

In this we can se that $25\times 25=\mathbf{6}25=\mathbf{(}\mathbf{2}\mathbf{\times }\mathbf{3}\mathbf{)}25$, $6$ is product of tens digit number and its successor.

In, $35\times 35=\mathbf{12}25=\mathbf{(}\mathbf{3}\mathbf{\times }\mathbf{4}\mathbf{)}25$, same pattern is observed.

So, using above pattern, we can answer

Hence, the missing values are

$$\begin{gathered} 55\times 55=(5\times 6)25=3025 \\ 65\times 65=(6\times 7)25=4225 \\ 75\times 75=(7\times 8)25=5625 \\ 85\times 85=(8\times 9)25=7225 \\ 95\times 95=(9\times 10)25=9025 \\ 105\times 105=(10\times 11)25=11025 \end{gathered}$$