Question

How do you find the remainder of \[{3^{983}}\] divided by \[5\].

Answer 1

Hint: In this question first, we find the pattern of the number \[3\]. After that, we write the last digit of the pattern and leave when repeated and then we find the multiple of a power of \[3\]. Then find the last digit of the power and the number which is divided by the power.

Complete step by step answer:
In this question, first, we find the power of 3.
\[3,\;9,\;27,\;81,\;243,\;729.............\]
We take the last digit from above the power of \[3\]pattern.
Hence the last digits are as below.
\[ \Rightarrow 3,9,7,1\left( {3,9.......} \right)\]
In the above pattern, it is seen that after the fourth value, the last digit is repeated.
Hence, the power of \[3\] that are multiples of \[4\]. And also, it is seen that all have the last digit one \[1\].
Then we check the power how much more.
\[ \Rightarrow 983 = 3 + 980\]
\[983\] Is more than a multiple of four \[(4) = 3\]
Therefore, the last digit of \[{3^{983}}\] indicates the third term in the sequence. And which is equal to \[7\].
All multiples of \[5\] end in either \[0\] or\[5\].
Now, we check \[7\] is closer to \[5\] and we find that \[7\] is closer to \[5\] comparison\[10\]. And it is \[2\] more than\[5\]
When integer \[7\] is divided by \[5\], the reminder will come\[2\].

Therefore, when \[{3^{983}}\] is divided by \[5\], the remainder will be \[2\].

Note: As we know that when a power of a number is much higher, then it is not an easy task to find the remainder when divided by an integer. So, we check the pattern and select the number according to the value after that divide.