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If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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5 00:00

Difficulty:   5% (low)

Question Stats: 86% (01:25) correct 14% (02:04) wrong based on 94 sessions

HideShow timer Statistics If K is a positive integer, what is the remainder when $$13^{4K+2} +8$$ is divided by 10 ?

A) 7
B) 4
C) 2
D) 1
E) 0

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Originally posted by dcummins on 10 Dec 2018, 20:03.
Last edited by Bunuel on 10 Dec 2018, 21:36, edited 2 times in total.
Edited the question.
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Joined: 02 Aug 2009
Posts: 7763
Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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1
1
hibobotamuss wrote:
I don't quite get this. Can someone explain a little better? Bunuel chetan2u

Hi...
When you divide any number by 10, the remainder is the units digit..
WHY?..
Because the closest multiple of 10 to that number will be the last number having units digit as 0. So whatever extra is there in units digit is the remainder.

So we have to find the units digit of the term.
All digits have a cylicity when it comes to units digit...
3 gives 3,9,7,1,3,9,7,1,3,9.....
$$3^1=3;3^2=9;3^3=27,3^4=81;3^5=243$$..unit digits are 3,9,7,1,3...
Also any number having 3 as units digit will always have same cylicity as 3, so let it be 1876543 or just 3, both will follow same cylicity...
Therefore $$13^{4k+2}$$ will have same units digit as 13^2, so units digit will be 9..
So the units digit of entire term will be 9+8=17, thus 7 will be the remainder.

A
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Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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Bunuel sir, Can you rectify the question? is it $$(13^{4K}+2)+8$$ or $$(13^4*K+2)+8$$
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Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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How the remainder is 7?
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Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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sghoshgt wrote:
How the remainder is 7?

sghoshgt, exactly, shouldn't the remainder be 1?

Posted from my mobile device
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If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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To solve this problem examine the functioning of the units digit of $$13^4^k+2$$

Any integer >10 we can refer to the units digit to identify the pattern for subsequent powers of that digit as follows:
$$13^1= 13$$

$$13^2 = 169$$

$$13^3 = 2197$$

$$13^4 = 28,561$$

This is the 3s units digit pattern
$$3-9-7-1$$

It has a cyclicity of 4, meaning its unit digit repeats each multiple of 4.

Thus units digit of 3-9-7-1-3-9 add 8 = 17
17 --> Units digit of 7
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Originally posted by dcummins on 10 Dec 2018, 20:55.
Last edited by dcummins on 12 Jun 2019, 14:24, edited 3 times in total.
Senior Manager  P
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Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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Afc0892 wrote:
Bunuel sir, Can you rectify the question? is it $$(13^{4K}+2)+8$$ or $$(13^4*K+2)+8$$

I've rectified it.

Apologies, I'm getting use to the forum code.
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Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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dcummins wrote:
To solve this problem examine the functioning of the units digit of $$13^4^k+2$$

Any integer >10 we can refer to the units digit to identify the pattern for subsequent powers of that digit as follows:
$$13^1= 13$$

$$13^2 = 169$$

$$13^3 = 2197$$

$$13^4 = 28,561$$

This is the 3s units digit pattern
$$3-9-7-1$$

It has a cyclicity of 4, meaning it repeats each multiple of 4.

Thus units digit of 3-9-7-1-3-9 add 8 = 17
1[/m]7 --> Units digit of 7

The question is 13 Whole to the power 4k+2 and not just $$13^{4k} + 2$$?
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Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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Afc0892 wrote:
dcummins wrote:
To solve this problem examine the functioning of the units digit of $$13^4^k+2$$

Any integer >10 we can refer to the units digit to identify the pattern for subsequent powers of that digit as follows:
$$13^1= 13$$

$$13^2 = 169$$

$$13^3 = 2197$$

$$13^4 = 28,561$$

This is the 3s units digit pattern
$$3-9-7-1$$

It has a cyclicity of 4, meaning it repeats each multiple of 4.

Thus units digit of 3-9-7-1-3-9 add 8 = 17
1[/m]7 --> Units digit of 7

The question is 13 Whole to the power 4k+2 and not just $$13^{4k} + 2$$?

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Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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I don't quite get this. Can someone explain a little better? Bunuel chetan2u
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Joined: 12 Sep 2017
Posts: 274
Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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Hi...
When you divide any number by 10, the remainder is the units digit..
WHY?..
Because the closest multiple of 10 to that number will be the last number having units digit as 0. So whatever extra is there in units digit is the remainder.

So we have to find the units digit of the term.
All digits have a cylicity when it comes to units digit...
3 gives 3,9,7,1,3,9,7,1,3,9.....
$$3^1=3;3^2=9;3^3=27,3^4=81;3^5=243$$..unit digits are 3,9,7,1,3...
Also any number having 3 as units digit will always have same cylicity as 3, so let it be 1876543 or just 3, both will follow same cylicity...
Therefore $$13^{4k+2}$$ will have same units digit as 13^2, so units digit will be 9..
So the units digit of entire term will be 9+8=17, thus 7 will be the remainder.

A[/quote]

I just have one question, how do we know that the cyclicity number that we have to choose is 9?

Kind regards!
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Joined: 02 Aug 2009
Posts: 7763
Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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1
jfranciscocuencag wrote:
Hi...
When you divide any number by 10, the remainder is the units digit..
WHY?..
Because the closest multiple of 10 to that number will be the last number having units digit as 0. So whatever extra is there in units digit is the remainder.

So we have to find the units digit of the term.
All digits have a cylicity when it comes to units digit...
3 gives 3,9,7,1,3,9,7,1,3,9.....
$$3^1=3;3^2=9;3^3=27,3^4=81;3^5=243$$..unit digits are 3,9,7,1,3...
Also any number having 3 as units digit will always have same cylicity as 3, so let it be 1876543 or just 3, both will follow same cylicity...
Therefore $$13^{4k+2}$$ will have same units digit as 13^2, so units digit will be 9..
So the units digit of entire term will be 9+8=17, thus 7 will be the remainder.

A

I just have one question, how do we know that the cyclicity number that we have to choose is 9?

Kind regards![/quote]

Hi ..
We know cylicity is 3,9,7,1,3,9,...
So the cylicity is in 4s...(3,9,7,1),(3,9,7,1)....
So the cylicity is same for first, fifth, ninth terms.... And for second, sixth, tenth and so on.
That is every fourth term is same...
Here we have 13^{4k+2}...
Now 4k means every fourth term, that is 13^{4k} will have _,_,_,1,_,_,_,1...that is it will have 1..
But we are looking for (4k+2)th term and it will have same number as the 2nd term, and, therefore, it is 9...
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Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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chetan2u wrote:
jfranciscocuencag wrote:
Hi...
When you divide any number by 10, the remainder is the units digit..
WHY?..
Because the closest multiple of 10 to that number will be the last number having units digit as 0. So whatever extra is there in units digit is the remainder.

So we have to find the units digit of the term.
All digits have a cylicity when it comes to units digit...
3 gives 3,9,7,1,3,9,7,1,3,9.....
$$3^1=3;3^2=9;3^3=27,3^4=81;3^5=243$$..unit digits are 3,9,7,1,3...
Also any number having 3 as units digit will always have same cylicity as 3, so let it be 1876543 or just 3, both will follow same cylicity...
Therefore $$13^{4k+2}$$ will have same units digit as 13^2, so units digit will be 9..
So the units digit of entire term will be 9+8=17, thus 7 will be the remainder.

A

I just have one question, how do we know that the cyclicity number that we have to choose is 9?

Kind regards!

Hi ..
We know cylicity is 3,9,7,1,3,9,...
So the cylicity is in 4s...(3,9,7,1),(3,9,7,1)....
So the cylicity is same for first, fifth, ninth terms.... And for second, sixth, tenth and so on.
That is every fourth term is same...
Here we have 13^{4k+2}...
Now 4k means every fourth term, that is 13^{4k} will have _,_,_,1,_,_,_,1...that is it will have 1..
But we are looking for (4k+2)th term and it will have same number as the 2nd term, and, therefore, it is 9...[/quote]

+Kudos

Thank you very much!!! So clear now.
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Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is  [#permalink]

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13^(4k+2)=13^4k.13^2
Since it is divided by 10 so we need to consider just the units digit.
13^4k (3,9,7,1-cyclicity) will always have the units digit as 1 and 13^2 has a units digit as 9.
So ((1*9)+8)/10=7

Posted from my mobile device Re: If K is a positive integer, what is the remainder when 13^(4K+2) +8 is   [#permalink] 12 Jun 2019, 10:38
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