How Cheenta works to ensure student success?
Explore the Back-Story
Problems and Solutions from CMI Entrance 2022.  Learn More 

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.

Some Useful Links

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.

Some Useful Links

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Knowledge Partner

Cheenta is a knowledge partner of Aditya Birla Education Academy
Cheenta

Cheenta Academy

Aditya Birla Education Academy

Aditya Birla Education Academy

Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.
JOIN TRIAL
support@cheenta.com
magic-wand