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


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