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If 3^10–n is divisible by 4, which of the following could be the value

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If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

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New post Updated on: 01 Aug 2018, 01:28
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If \(3^{10}–n\) is divisible by 4, which of the following could be the value of an integer n?

I. 0
II. 1
III. 5

A. I only
B. II only
C. III only
D. II and III only
E. I, II and III

Originally posted by sharank on 10 Jul 2018, 15:28.
Last edited by Bunuel on 01 Aug 2018, 01:28, edited 3 times in total.
Added the topic name and formatted the question
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Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

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New post 10 Jul 2018, 18:58
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sharank wrote:
) If 3^10–n is divisible by 4, which of the following could be the value of an integer n?
I. 0 II. 1 III. 5

A. I only B. II only C. III only D. II and III only E. I, II and III

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1) choices
The choices give you the answer..
3^10-n
I. If n is 0, we are left with 3^10, whi h is ODD, so not div by 4
Ii. Now 1 and 5 have a difference of 4 within themselves so either both are possible or none of the two

In choices D gives both and there is no choice giving None of the above
So D


2) cyclicity
3 gives a remainder 3
3^2=9 gives a remainder 1
3^3 =27 gives a remainder 3
3^4 =81 gives a remainder 1..
So even power leave a remainder 1
So 3^10 will leave a remainder 1 so n can be 1
Now 5 also leaves a remainder 1 so 5 can also be the answer..
D
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Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

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New post 10 Jul 2018, 19:34
sharank wrote:
If 3^10–n is divisible by 4, which of the following could be the value of an integer n?



Is it \(3^{10-n}\) or \(3^{10}-n\)
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Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

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New post 10 Jul 2018, 19:35
chetan2u could you tell what is wrong with my reasoning.

3 power 10 would end in 0 - 5 would give another odd number. Hence only divisive by n=1

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Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

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New post 10 Jul 2018, 19:48
rever08 wrote:
chetan2u could you tell what is wrong with my reasoning.

3 power 10 would end in 0 - 5 would give another odd number. Hence only divisive by n=1

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3*10 would end in 0
But 3^10=3*3*3*3*...10times
Cyclicity
3^1 ends in 3
3^2 =9 ends in 9
3^3 = 27 ends in 7
3^4=27*3 ends in 1
3^5=81*3 ends in 3 and the cyclicity happens after every 4th number
So 3^10=3^(4*2+2) so will have same units digit as 3^2 so 9..
So 3^anything positive will always be ODD
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Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

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New post 10 Jul 2018, 19:51
chetan2u wrote:
rever08 wrote:
chetan2u could you tell what is wrong with my reasoning.

3 power 10 would end in 0 - 5 would give another odd number. Hence only divisive by n=1

Posted from my mobile device


3*10 would end in 0
But 3^10=3*3*3*3*...10times
Cyclicity
3^1 ends in 3
3^2 =9 ends in 9
3^3 = 27 ends in 7
3^4=27*3 ends in 1
3^5=81*3 ends in 3 and the cyclicity happens after every 4th number
So 3^10=3^(4*2+2) so will have same units digit as 3^2 so 9..
So 3^anything positive will always be ODD


Argh..what was I thinking??
Thanks mate.
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Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

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New post 10 Jul 2018, 21:54
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sharank wrote:
If \(3^{10}–n\) is divisible by 4, which of the following could be the value of an integer n?
I. 0
II. 1
III. 5

A. I only
B. II only
C. III only
D. II and III only
E. I, II and III

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Given \(3^{10}–n\) = \(4k\)

I. \(n = 0\), we get \(3^{10}–n\) = \(3^{10}\) = \(3^{4}*3^{4}*3^{2}\)

When \(3^{4}*3^{4}*3^{2}\) is divided by \(4\), we get remainder as \(1\). Hence Not Divisible.

\(n = 0\) is not possible.


II. \(n = 1\), we get \(3^{10}–n\) = \(3^{10}–1\)

we know from above \(3^{10}\) divided by \(4\) leaves a remainder of \(1\)

& \(1\) divided by \(4\) will leave a remainder of \(1\).

Hence \(3^{10}–1\) divided by 4 will leave a remainder \((1 - 1) = 0\). Hence divisible.

\(n = 1\) is possible.


III. \(n = 5\), we get \(3^{10}–n\) = \(3^{10}–5\)

we know from above \(3^{10}\) divided by \(4\) leaves a remainder of \(1\)

& \(5\) divided by \(4\) will leave a remainder of \(1\).

Hence \(3^{10}–5\) divided by 4 will leave a remainder \((1 - 1) = 0\). Hence divisible.

\(n = 5\) is possible.


Answer D.


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Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

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New post 19 Jul 2018, 09:29
If you don't subtract anything than 3^10-0 will be odd hence option 1 and 5 can be eliminated immediately.

Now 3^10 is a big number, say x. If either 1 or 5 is subtracted from a big number the both the numbers will be divisible by 4 if either one of them is divisible by 4. (e.g. 9-1 = 8 & 9-5=4; BOTH 8 & 4 are divisible by 4)

Since there is no option which says none of these then we can safely select option D with doing any calculations!! Hope this out of the box thinking was helpful & time saving :-)
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Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

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New post 01 Aug 2018, 01:26
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3^10 - n/ 4 = Q

So 3's cycle of power is 3,9,7,1...if we raise 3 to 10 times we will get a number that ends in XX9.
So 9 - 5 = 4 = divisible
9 - 8 = divisible.

So answer is II,III which is D
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Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

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New post 09 Aug 2018, 04:38
3=3
3*3 =9
3*3*3=27
3*3*3*3=81

then the last digit repeats itself.
so starting from 3 count 10 times we come to 9(last digit)

then we know that the last digit for 3^10 is 9.
and this -n is divisible by 4.that means n will be 1 or 5.
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Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

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New post 10 Sep 2018, 00:50
chetan2u wrote:
rever08 wrote:
chetan2u could you tell what is wrong with my reasoning.

3 power 10 would end in 0 - 5 would give another odd number. Hence only divisive by n=1

Posted from my mobile device


3*10 would end in 0
But 3^10=3*3*3*3*...10times
Cyclicity
3^1 ends in 3
3^2 =9 ends in 9
3^3 = 27 ends in 7
3^4=27*3 ends in 1
3^5=81*3 ends in 3 and the cyclicity happens after every 4th number
So 3^10=3^(4*2+2) so will have same units digit as 3^2 so 9..
So 3^anything positive will always be ODD


can we reason as follows:

since 3^10 is odd, 3^10 -1 is even (odd-odd gives even). and since we get an even result, it could be divisible either by 4 or 5 ?

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Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

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New post 10 Sep 2018, 15:41
3^10=729^2= 29*29 = a number ending with 41. Therefore:
1) 41-0 = 41 not divisible by 4
2) 41-1 = 40 divisible by 4
3) 41-5 = 36 divisible by 4
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Re: If 3^10–n is divisible by 4, which of the following could be the value  [#permalink]

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New post 10 Sep 2018, 16:30
3^10 = (4-1)^10 = 4k+1, where k is an integer.
So any n = 4m+1 will work, where m is an integer. Only 1 and 5 can be represented as 4m+1.

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Re: If 3^10–n is divisible by 4, which of the following could be the value   [#permalink] 10 Sep 2018, 16:30
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