Question

If the unit's digit of \[{7^3}\] is \[3\] then what will be the unit’s digit of \[{7^{11}}\].
a). \[7\]
b). \[9\]
c). \[3\]
d). \[1\]

Answer 1

Hint: When we find the unit digit of a number we see what we get when we multiply unit digits of two numbers that give us our desired number. As we are given with the unit digit of the number \[{7^3}\] that is equal to \[3\]. So we factorize our desired number in powers of \[3\] and compute the result.

Complete step-by-step solution:
Given, that the unit digit if \[{7^3}\] is equal to \[3\]. Let us factorize the number \[{7^{11}}\] into powers \[3\] and then compute the result.
\[{7^{11}} = {7^3}{.7^3}{.7^3}{.7^2}\]
Therefore, unit digit of \[{7^{11}}\]= (unit digit of\[{7^3}\]) x (unit digit of\[{7^3}\]) x (unit digit of\[{7^3}\]) x (unit digit of\[{7^2}\])
\[{7^2} = 7.7 = 49\]
So, the unit digit of\[{7^2}\] is \[9\].
Then, unit digit of \[{7^{11}}\]= (\[3\]) \[ \times \] (\[3\]) \[ \times \] (\[3\]) \[ \times \](\[9\]), which is equal to \[243\].
Hence, the unit digit of \[{7^{11}}\]= unit digit of \[243 = \]\[3\]
Therefore, the unit digit of \[{7^{11}}\]\[ = \]\[3\]
Hence, the correct option is (c)\[3\].
Additional information: The units’ digit of the product of any two given numbers is the same as the units’ digit of the two numbers’ units’ digits. Hence it is very important that when we factorize the number we factorize in such a way that the units’ digit of the terms of the factorized numbers is known to us.

Note: When we proceed with the computation of \[{7^{11}}\] it is very important that we approach with a strategic process which is that we use the result given that the unit digit of \[{7^3}\]is \[3\]. Thus in this manner, the computation will be feasible and will be easier for us.