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What is the units digit of 2^39?

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What is the units digit of 2^39?  [#permalink]

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New post 25 Feb 2017, 19:35
1
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Question Stats:

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What is the units digit of 2^39?

A) 2
(B) 4
(C) 6
(D) 8
(E) 9
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Re: What is the units digit of 2^39?  [#permalink]

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New post 25 Feb 2017, 23:03
vikasp99 wrote:
What is the units digit of 2^39?

A) 2
(B) 4
(C) 6
(D) 8
(E) 9


\(2^1 = 2\)
\(2^2 = 4\)
\(2^3 = 8\)
\(2^4 = 16\)

\(2^5 = 32\)


Thus, the cyclicity of units digit 2 is 4...


Now, \(2^{39} = 2^{36}*2^3\)

Or, \(2^{39} = 2^{4*9}*2^3\)

\(2^{39} = 6*8\) ( As 2^4 = Units digit 6 ) & ( 2^3 = Units digit 8 )

Thus, the units digit of 2^{39} = 8

Hence, the answer must be (D) 8
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What is the units digit of 2^39?  [#permalink]

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New post 10 Mar 2018, 14:38
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vikasp99 wrote:
What is the units digit of 2^39?

A) 2
(B) 4
(C) 6
(D) 8
(E) 9

To see the theory behind cyclicity, go to LAST DIGIT OF A POWER

As Abhishek009 notes, to find the units digit of an integer raised to a power, find the pattern the integer follows when raised to increasing powers.

Watch only the units digits.

The number of units digits in one iteration of the pattern is "cyclicity."

\(2^1 = 2\)
\(2^2 = 4\)
\(2^3 = 8\)
\(2^4 =16\)

\(2^5 = 3[2]\)
\(2^6 = 6[4]\)
\(2^7 = ...8\)
\(2^2 = ...6\)


Every four powers, the units digits have a pattern of 2, 4, 8, 6

Units digit of \(2^{39}\)?
Cyclicity is 4.
Divide the exponent by that cyclicity of 4.
\(\frac{39}{4} = 9\) . . .
with remainder, \(r = 3\)

Remainder \(r\) gives you the units digit you are looking for*
r = 3? The huge number in the prompt has the same units digit as \(2^3\)

\(2^3\) has a units digit of \(8\)

Answer D

*If r = 2, your huge number's units digit is the same as that of \(2^2\), which is 4.

If remainder r = 1, the units digit of the unknown number is the same as \(2^1\), which is 2.

If there is no remainder, r = 0, your huge number's units digit is the same as the cyclicity's number.
If this prompt asked for the units digit of \(2^{40}\), e.g., the remainder would be 0. (Exponent/Cyclicity = 40/4 = 10). If r = 0, the units digit will be the same as \(2^4\)
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What is the units digit of 2^39?  [#permalink]

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New post 10 Mar 2018, 19:24
Hi
To solve these types of questions, it is not necessary to remember cyclicity order of different digits,
In fact, all digits follow a common cyclicity order, they repeat itself after 4k+1 power.
Steps to solve such questions:
1) divide the power by 4 and find remainder.
here the remainder is 3
2) Now find the unit digit by raising it to exponent of remainder.(if remainder is 0, raise it to exponent 4)
here, it is 2^3 = 8

this method works for every digit
:cool:

(PS: for some of digits , we have simpler pattern method.
1) 0 - always 0
2) 4 - odd power = 4, even power = 6
3) 5 always 5
4) 6 always 6
5) 9 - odd power 9, even power = 1)
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