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Last digit of \(97^{2013}\) (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.

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