What is the unit digit of 7^86?

Question

(A) 0
(B) 1
(C) 3
(D) 7
(E) 9

LaveenaPanchal · Answer

The Cycle for 7^ is 4

1st Power - 7
2nd Power -9, 
3rd Power - 3
4th Power - 1

86/4 gives a remainder of 2. so the units digit will be 2nd power i.e. 9. Option E

Nidzo · Answer

7 follows the four number pattern of: 7, 9, 3, 1.

86/4 = 21 with a remainder of 2, so the answer will be 9.

Answer E

MathRevolution · Answer

7^{86} = (7^4)^{21} * 7^2

=>  7^2 = 49

Thus, unit digit is 9.

Answer E

CrackverbalGMAT · Answer

Solution:

To find unit digit, you have to compute the Cyclicity of 7

Check the cycle for 7 which is mentioned here-

7^1-------> Unit digit 7

7^2--------> Unit digit 9

7^3-------->Unit digit 3

7^4 -------> Unit digit 1

7^5--------> Unit digit 7=>The cycle repeats after 4 steps

There are 86 7s and 86 on dividing with 4 leaves a remainder of 2

=> Unit digit is 9  (option e)

Devmitra Sen
GMAT SME

StenAnsiktet · Answer

I struggle a lot with these kinds of questions so I am asking you all for help:

I do not know the cyclicity of numbers by heart, and to me the task of computing 7^3 and 7^4 is extremely time-consuming. 
Is there any trick to apply to get the cyclicity in a less arithmetic-heavy way? Or do I have to work on my arithmetic to brute-force it?

Very interested to hear your input!

Bunuel · Answer

Theory is here: https://gmatclub.com/forum/math-number- ... 88376.html Check Units digits, exponents, remainders problems directory in our Special Questions Directory . Hope it helps.

StenAnsiktet · Answer

Thanks as always Bunuel! 
I’ve gotta say that page on number theory is extremely condensed, must have been tough chipping away at the explanations in order to reach this reduced form, good job!

Posted from my mobile device

BrushMyQuant · Answer

What is the units digit of  7^{86} ?

Now to find the unit's digit of  7^{86} , we need to find the pattern / cycle of unit's digit of power of 7 and then generalizing it.

Unit's digit of  7^1  = 7
Unit's digit of  7^2  = 9
Unit's digit of  7^3  = 3
Unit's digit of  7^4  = 1
Unit's digit of  7^5  = 7

So, unit's digit of power of 7 repeats after every  4^{th}  number.
=> We need to divided 86 by 4 and check what is the remainder
=> 86 divided by 4 gives 2 remainder

=>  7^{86}  will have the same unit's digit as  7^2 
=> Unit's digits of  7^{86}  = 9

So,  Answer will be E 
Hope it helps!

Link to Theory for Last Two digits of exponents here    .

Link to Theory for Units' digit of exponents here    .

What is the unit digit of 7^86?

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What is the unit digit of 7^86?

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