Question

What is the last digit in the expansion of \[{(2457)^{754}}\]?
a). \[3\]
b). \[7\]
c). \[8\]
d). \[9\]

Answer 1

Hint: Here, in this question, we have to find the last digit of \[{(2457)^{754}}\]. So, for finding the last digit of the given exponent we have to find the pattern in the expansion of \[{(2457)^{754}}\] and the last digit of the number \[{(2457)^{754}}\] is same as the last digit of \[{7^{754}}\] because this number ends with \[7\]. Now find the pattern expansion of \[{7^{754}}\].

Complete step-by-step solution:
Given,
\[{(2457)^{754}}\]
To find,
The last digit in the expansion of \[{(2457)^{754}}\].
Last digit in the expansion of \[{(2457)^{754}}\] is same as last digit in the expansion of \[{7^{754}}\] because the unit place of the number \[2457\] is \[7\].
So, now we have to find the last digit in the expansion of \[{7^{754}}\].
The pattern observed in the expansion of \[{7^{754}}\]

Number including the power	The number after calculating the power	Unit digit of the calculated number
\[{7^1}\]	\[7\]	\[7\]
\[{7^2}\]	\[49\]	\[9\]
\[{7^3}\]	\[553\]	\[3\]
\[{7^4}\]	\[3871\]	\[1\]
\[{7^5}\]	\[27097\]	7

Here, the pattern is observed that after every fourth power again the unit digit is \[7\].
So, if the power is multiple of \[4\] then the unit term of that number is 1
Writing power of \[{7^{754}}\] in multiplication of \[4\].
\[{7^{754}} = {7^{752}} \times {7^2}\]
\[752\] is a multiple of \[4\] so the unit digit is \[1\]
So, the last digit of the number \[{7^{754}}\] is the same as the last digit of the number\[{7^2}\].
We know that \[{7^2} = 49\]
So, last digit of the number \[{7^2}\] is \[9\]
The last digit of the number \[{(2457)^{754}}\] is
\[ \Rightarrow 9\]
Final answer:
The last digit of the number \[{(2457)^{754}}\] is:
 \[ \Rightarrow 9\]
Option d is the correct answer.

Note: Here, in this question, we have to use the trick that the last digit of any number having power is the same as the last digit of the unit-placed number having the same power. For further calculation, we have to find the pattern that is repeating after a particular number. In this case, the pattern is found after \[4th\] power.