Let's discuss a problem from TIFR 2014 Problem 18.
Question:
What is the last digit of (97^{2013})?
Discussion:
(97 \equiv -3 (\mod 10 ) )
(97^2 \equiv (-3)^2 \equiv -1 (\mod 10 ) )
(97^3 \equiv (-1)\times (-3) \equiv 3 (\mod 10 ) )
(97^4 \equiv (3)\times (-3) \equiv 1 (\mod 10 ) ).
Now, (2013=4\times 503 +1).
(97^{4\times 503+1} \equiv (1^{503})\times (97) \equiv 7 (\mod 10 ) ).
So the last digit of (97^{2013}) is 7.
Let's discuss a problem from TIFR 2014 Problem 18.
Question:
What is the last digit of (97^{2013})?
Discussion:
(97 \equiv -3 (\mod 10 ) )
(97^2 \equiv (-3)^2 \equiv -1 (\mod 10 ) )
(97^3 \equiv (-1)\times (-3) \equiv 3 (\mod 10 ) )
(97^4 \equiv (3)\times (-3) \equiv 1 (\mod 10 ) ).
Now, (2013=4\times 503 +1).
(97^{4\times 503+1} \equiv (1^{503})\times (97) \equiv 7 (\mod 10 ) ).
So the last digit of (97^{2013}) is 7.
