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# What is the units digit of 2^39?

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Senior Manager
Joined: 02 Jan 2017
Posts: 310
What is the units digit of 2^39?  [#permalink]

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25 Feb 2017, 19:35
1
3
00:00

Difficulty:

15% (low)

Question Stats:

74% (00:35) correct 26% (00:47) wrong based on 78 sessions

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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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Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4276
Location: India
GPA: 3.5
Re: What is the units digit of 2^39?  [#permalink]

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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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Abhishek....

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Senior SC Moderator
Joined: 22 May 2016
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What is the units digit of 2^39?  [#permalink]

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10 Mar 2018, 14:38
1
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 $$8$$

*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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Joined: 27 Oct 2017
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What is the units digit of 2^39?  [#permalink]

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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

(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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What is the units digit of 2^39? &nbs [#permalink] 10 Mar 2018, 19:24
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