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Last digit of \(97^{2013}\) (TIFR 2014 problem 18)

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.

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