Lesson: Units Digit of Large Powers

Comment on Units Digit of Large Powers


Could you elaborate on more complicated cycles? (n > 3) I'm still confused on how to find the units digit in those situations after I find the cycle and cycle length.
gmat-admin's picture

Sure thing.
Let's try the units digit of 7^33

First find the pattern AND the cycle:
7^1 = 7
7^2 = 49
7^3 = --3
7^4 = --1
7^5 = --7
7^6 = --9
7^7 = --3
7^8 = --1
So, the CYCLE = 4

Now focus on the MULTIPLES OF 4 (the cycle)
7^4 = ---1
7^8 = ---1
7^12 = ---1

We want the units digit of 7^33
Since 32 is a multiple of 4, we know that 7^32 = ---1
Since the cycle is 7, 9, 3, 1, 7, 9, 3, 1, etc, we know that the NEXT POWER, 7^33, has units digit 7

Does that help?


sir in questions like 234^121
here sir can we make pattern of unit digit(by finding cycle) of 4^121 rather than 234^121 since we are interested only in unit digit?
gmat-admin's picture

Yes, that's correct. The units digit of 4^121 will be the same as the units digit of 234^121.


ari.banerjee's picture

Hi Brent,

Is it safe to assume that the cycle ends as soon as you get 1?

Because any no: multiplied by 1 will repeat that units digit?

So I start looking for pattern and as soon as I reach 1 i count the no:?

Thank you,
Ari Banerjee
gmat-admin's picture

Yes, that rule will work for powers of integers ending in 1, 3, 7 and 9


Hi Brent,

In one of the questions https://gmatclub.com/forum/what-is-the-units-digit-of-the-solution-to-217680.html

You have advised to quickly reach a conclusion as follows:

"These questions can be time-consuming. If you're pressed for time, you can use the following approach to reduce the answer choices to just 2 options in about 5 seconds.

177^(28) - 133^(23) = (odd number)^(some positive integer) - (odd number)^(some positive integer)
= odd - odd"

Please can you advise what are the other rules that we can memorise. e.g. Odd number, even integer etc.

Not sure if there is a video for that later on.


gmat-admin's picture

You can find additional rules regarding odd and even integers here: https://www.gmatprepnow.com/module/gmat-integer-properties/video/837

Cheers, Brent

Ok, Thanks Brent.

Hi Brent, thanks for this course. I have a question to this question from the Official GMAT book 2021:

If 2^x + y = 48, what is the value of y ?

1) x^2 = 81
2) x − y = 2

I thought it would be "both statements ALONE are sufficient" as you can take the square root of x in 1) and thereby find y, and in 2) you can substitute x with 2-y to find y.

However, the solution is that statement 2 is sufficient but 1 is not.

Please help! Again, thanks a lot.

gmat-admin's picture

Be careful; there are TWO solutions to the equation x² = 81.
Either x = 9, or x = -9
When we plug both values into the equation 2^x + y = 48, we get TWO possible values for y.
So, statement 1 is not sufficient

.. of course!! Thanks!

Could you please help with this one? My answer is 4 as both numbers have 7 as units digit according to their cycles so don’t understand why the answer is zero.

I've noticed that a huge number of the reinforcement questions in the root section so far are all in the mid to high range of difficulty. Is this because if we get a question in the exam about roots it is probably in the medium to high range of difficulty? Are there very little or no easy root questions?
gmat-admin's picture

That's correct. A lot of students struggle with questions involving roots. So, there aren't many roots questions rated as easy.

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