Sequences & Patterns - Source from Manhattan GMATPREP

[shanice]

If Sn = 3^n, what is the units digit of S65?

Clearly, you cannot be expected to multiply out 365 on the GMAT. Therefore, you must
assume that there is a pattern in the powers of three.

3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
3^6 = 729
3^7 = 2,187
3^8 = 6,561

Note the pattern of the units digits in the powers of 3: 3, 9, 7, 1,
[repeating] ... Also note that the units digit of Sn,when n is a multiple
of 4, is always equal to 1. You can use the multiples of 4 as
"anchor points" in the pattern. Since 65 is 1 more than 64 (the closest
multiple of 4), the units digit of S65 will be 3, which always follows
1 in the pattern.

Can anyone explain as I don't understand the explanation from the book.

Thank you.

[Anurag@Gurome]

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]

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?

Thank you.

[shanice]

Sir, can I know whether my understanding is correct as posted earlier?

Thank you in advance.

[Anurag@Gurome]

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?"

[shanice]

Hello Sir,

I'm extremely sorry for the late reply for the below example:-

What is the unit's digit of 23^23?"

23^0=1
23^2=23
23^3=529
23^4=12167
23^5=279841
23^6=6436343
23^7=148035889

There is a pattern in the unit's digit of the powers of 23 - 1,3,9,7,1,3,9,7,1,....

The power of 23 is a multiple of 5 with 1 as a unit's digit. Therefore, 23^23=23^(20+3)=(23^20)*(23).
20 is a multiple of 5 so the unit's digit of 23^23 should be 3.

Am I correct?