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If n is a positive integer, what is the remainder when

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Joined: 06 Apr 2010
Posts: 113
If n is a positive integer, what is the remainder when  [#permalink]

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23 Jun 2010, 11:35
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If n is a positive integer, what is the remainder when $$3^{8n+3}+2$$ is divided by 5?

A. 0
B. 1
C. 2
D. 3
E. 4
Math Expert
Joined: 02 Sep 2009
Posts: 53066
Re: PS Questions from SET 1-Need detail solution  [#permalink]

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23 Jun 2010, 13:40
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udaymathapati wrote:
Some difficult PS Questions from SET1. Please help me with resolving them...OAs have mentioned at the end.

Q13:
If n is a positive integer, what is the remainder when $$3^{8n+3}+2$$ is divided by 5?
A. 0
B. 1
C. 2
D. 3
E. 4

3 in power has cyclicity of 4:
1. 3^1=3 (last digit is 3)
2. 3^2=9 (last digit is 9)
3. 3^3=27 (last digit is 7)
4. 3^4=81 (last digit is 1)

5. 3^5=243 (last digit is 3 again!)
...

To find the last digit of $$3^{8n+3}$$, divide the power (which is $$8n+3$$) by cyclicity # ( which is $$4$$) and look at the remainder --> $$\frac{8n+3}{4}$$ --> $$remainder=3$$, which means that the last digit of $$3^{8n+3}$$ will be the same as the last digit of $$3^3=27$$ (last digit is 7). (Side note: If the remainder were 0, then last digit would be the same as the las digit of $$3^4$$).

Now, last digit of $$3^{8n+3}+2$$ will be $$7+2=9$$. Any integer with last digit 9 upon division by 5 yields remainder of 4.

Hope it's clear.
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Re: PS Questions from SET 1-Need detail solution  [#permalink]

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23 Jun 2010, 20:35
Bunuel wrote:
udaymathapati wrote:
Some difficult PS Questions from SET1. Please help me with resolving them...OAs have mentioned at the end.

Q13:
If n is a positive integer, what is the remainder when $$3^{8n+3}+2$$ is divided by 5?
A. 0
B. 1
C. 2
D. 3
E. 4

3 in power has cyclicity of 4:
1. 3^1=3 (last digit is 3)
2. 3^2=9 (last digit is 9)
3. 3^3=27 (last digit is 7)
4. 3^4=81 (last digit is 1)

5. 3^5=243 (last digit is 3 again!)
...

To find the last digit of $$3^{8n+3}$$, divide the power (which is $$8n+3$$) by cyclicity # ( which is $$4$$) and look at the remainder --> $$\frac{8n+3}{4}$$ --> $$remainder=3$$, which means that the last digit of $$3^{8n+3}$$ will be the same as the last digit of $$3^3=27$$ (last digit is 7). (Side note: If the remainder were 0, then last digit would be the same as the las digit of $$3^4$$).

Now, last digit of $$3^{8n+3}+2$$ will be $$7+2=9$$. Any integer with last digit 9 upon division by 5 yields remainder of 4.

Hope it's clear.

Bunuel, : If the remainder were 0, then last digit would be the same as the las digit of 3^4=81 means 1.
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Joined: 02 Sep 2009
Posts: 53066
Re: PS Questions from SET 1-Need detail solution  [#permalink]

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23 Jun 2010, 21:01
1
udaymathapati wrote:

Bunuel, : If the remainder were 0, then last digit would be the same as the las digit of 3^4=81 means 1.

No, as in this case all numbers in power which is multiple of cyclicity would have the last digit of 1.

When power is divisible by cyclicity # (remainder 0), then the last digit is the same as the last digit of number in the power of cyclicity #.

For example:
Cyclicity of 2 is 4, so $$2^{4n}$$ (where $$n$$ is a positive integer) will have the same last digit as $$2^4$$, which is 6.

OR:

Cyclicity of 4 is 2, so $$4^{2n}$$ (where $$n$$ is a positive integer) will have the same last digit as $$4^2$$, which is 6.

Hope it's clear.
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Joined: 06 Apr 2010
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Re: PS Questions from SET 1-Need detail solution  [#permalink]

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24 Jun 2010, 11:08
1
Bunuel wrote:
udaymathapati wrote:

Bunuel, : If the remainder were 0, then last digit would be the same as the las digit of 3^4=81 means 1.

No, as in this case all numbers in power which is multiple of cyclicity would have the last digit of 1.

When power is divisible by cyclicity # (remainder 0), then the last digit is the same as the last digit of number in the power of cyclicity #.

For example:
Cyclicity of 2 is 4, so $$2^{4n}$$ (where $$n$$ is a positive integer) will have the same last digit as $$2^4$$, which is 6.

OR:

Cyclicity of 4 is 2, so $$4^{2n}$$ (where $$n$$ is a positive integer) will have the same last digit as $$4^2$$, which is 6.

Hope it's clear.

Thanks bunuel. It's clear for me.
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Joined: 22 Jun 2010
Posts: 2
Re: PS Questions from SET 1-Need detail solution  [#permalink]

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25 Jun 2010, 20:03
Ok, beautiful solution, but to time consuming...
It´s a problem solving question, so, there is just one answer right?
Just to n = 1 and make the calculation...
Math Expert
Joined: 02 Sep 2009
Posts: 53066
Re: If n is a positive integer, what is the remainder when  [#permalink]

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09 Mar 2014, 12:09
Bumping for review and further discussion.

For more on this kind of questions check Units digits, exponents, remainders problems collection.
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GMAT 1: 650 Q45 V31
GPA: 4
Re: If n is a positive integer, what is the remainder when  [#permalink]

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21 Aug 2017, 09:57
600 Level Question.
Took me 1 minute to solve.
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Joined: 07 Dec 2014
Posts: 1154
If n is a positive integer, what is the remainder when  [#permalink]

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21 Aug 2017, 11:36
udaymathapati wrote:
If n is a positive integer, what is the remainder when $$3^{8n+3}+2$$ is divided by 5?

A. 0
B. 1
C. 2
D. 3
E. 4

let n=1
check the units digits of 3,9,27,81
3^11, or 3^3+4+4, falls into the 3rd position in a cycle of 4, giving a units digit of 7
7+2=9
9/5 gives a remainder of 4
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Location: Kuwait
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WE: Engineering (Real Estate)
If n is a positive integer, what is the remainder when  [#permalink]

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27 Sep 2018, 23:57
3^(8n+3) this means it will always be the third cycle.

3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81 and the cycle repeats.

Since we know 3^(8n+3) will have a units digit like the one in 3^3

Then we can do 7+2 = 9/5 = 1 4/5

Or similarly we can take 27+2 = 29/5 = 5 4/5

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Joined: 15 Aug 2018
Posts: 51
GMAT 1: 740 Q47 V45
GPA: 3.5
Re: If n is a positive integer, what is the remainder when  [#permalink]

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20 Jan 2019, 06:55
Yea, Cyclicity seems to do the trick very well!
Doesn't this approach work, too, Bunuel ?:

(3^(8n+3)+2)/5

((5-2)^(8n+3)+2)/5

Let's now consider the binomial theorem. If applied here, we can see that 2 will get assigned to an uneven power of 2.
So this must be the remainder within the brackets. Adding the "+2" from outside the brackets, we get 4.

Is that fine?

Best, gota900
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Re: If n is a positive integer, what is the remainder when   [#permalink] 20 Jan 2019, 06:55
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