Sequences & Patterns - Source from Manhattan GMATPREP

Not sure if it can explained more clearly.
But let me give it a try.

3^0 = 1
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81

Form the above list, you can see that there is pattern in the unit's digit of powers of 3. The unit's digit of 3^4 is 1, Hence, unit's digit of 3^5 will be 3, as you are multiplying 3^4 with another 3. Then again multiplying by 3, you will get unit's digit (3*3) = 9...

Hence, the pattern of unit's digit of powers of 3 are 1, 3, 9, 7, 1, 3, 9, 7, 1, 3...

Now, note that whenever the power of 3 is a multiple of 4, the unit's digit is 1.

Hence, we can write 3^65 = 3^(64 + 1) = (3^64)*(3)
As 64 is a multiple of 4, the unit's digit of 3^64 will be 1.

Hence, 3^65 = 3^(64 + 1) = (3^64)*(3) = (Some big number with unit's digit 1)*(3) = (Some big number with unit's digit 1*3 = 3)

Hence, unit's digit of 3^65 is 3.

Hope that helps.

shanice wrote:Sir,after reading your explanation, can I conclude as below:-

The pattern of the units digits in the powers of 3 keeps on repeating as 1,3,9,7,1,3,9,7,1,....
In 3^4, the nearest for S65 is in multiple of 4 so :

4^1 = 4
4^2 = 16
4^3 = 64

Therefore, the answer is 3 because in the powers of 4 with a multiple of 3 S65 is the nearest to 64.Is my understanding correct? Can I use this way to do other questions?

Multiple of 4 and powers of 4 are not same.
You do not need to bring powers of 4 into picture.

I think I did not get what you are trying to say here. Can you explain your logic again? If possible with a different example, say "What is the unit's digit of 23^23?"