If r is a positive integer, then what is the remainder when 138

Question

If r is a positive integer, then what is the remainder when  138^{13+4r}  is divided by 10?

A. 1
B. 2
C. 4
D. 6
E. 8

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AUG1523 · Answer

IMO 4. I just found the units place of 8^14 with r=1. 8 has cyclicity of 4 (8 4 2 6), and 14 is 4*3+2 so the units digit is 4

tanyas311 · Answer

Given:  138^{13r+4}

=>  138^{12r+r+4} 
=>  138^{12r+4}*138^r 
Units digit of underlined expression will be 6 since the cyclicity of 8 is 4 (8, 4, 2, 6)
=>  6*138^r 
For r = 1, units digit would be 8 (6*8=48)
Therefore, the remainder will be 8.

Answer :   Option E

gmatophobia · Answer

Is there a reason you have assumed r = 1 ?

I agree that if r = 1, the remainder is 8. But 'r' can also be any other number, right ?

For example it can assume any of the below form -

r = 4k , the remainder is 6
r = 4k+1 , the remainder is 8
r = 4k+2 , the remainder is 4
r = 4k+3 , the remainder is 2

IMO the remainder can be all the options except for option A (ie. remainder cannot be 1). Am I missing anything ?

chetan2u   carcass  - Can you please share an OA to this question.

chetan2u · Answer

gmatophobia , you are correct in your observation.

As we are looking for remainder when divided by 10, the units digit will be the answer.

138^{13r+4}=138^{12r+r+4}=138^{4(3r+1)+r} 
So, the units digit will be same as the units digit of  138^r , and this will depend on value of r.

Incomplete or faulty question. It could be 13+4r instead of 13r+4

Bunuel · Answer

carcass  please edit the question. Thank you!

Carcass · Answer

OE

Given: r is a positive integer  ⟹ r > 0

If we come to know the unit digit of  138^{13r+4} , it is nothing but the remainder when  138^{13r+4}  is divided by 10. Also use plugging in.

Plug in r = 1

⟹ 138^{13+4} = 138^{17}

It is same as working with 8^{17}

So now, the question has reduced to, what is the remainder when 8^{17} is divided by 10.

If we find the unit digit of 8^{17}, we will come to know the reminder 817 is divided by 10.

Unit digit of 8^1 is 8
Unit digit of 8^2 is 4
Unit digit of 8^3 is 2
Unit digit of 8^4 is 6
Unit digit of 8^5 is again 8.

⟹ The unit digit will repeat after every four values that means, the cyclicity is 4. 
⟹ For8^16, unit digit is 6, as 16 is divisible by the cyclicity 4.
⟹ For 8^17, unit digit is 8.

Therefore, any number ending with 8, when divided by 10 gives the remainder as 8. 
Hence, the answer is E.

chetan2u · Answer

carcass , can you please share the source. The solution is not correct.

We do not know r. 
Plug as 1, then  138^{13*1+4}=138^{17}  same as 138^1 so the answer is 8.
Plug as 2, then  138^{13*2+4}=138^{30}  same as 138^2 so the answer is 8*8 or 4.
And so on.

Bunuel · Answer

Above is wrong.

If r = 1, then  138^{13r+4}=2387505984309882282990747828196343808  --> remainder upon division by 10 is 8;

If r = 2, then  138^{13r+4}=...138978614247424  --> remainder upon division by 10 is 4;

If r = 3, then  138^{13r+4}=...7788547900042348863550390272  --> remainder upon division by 10 is 2;

If r = 4, then  138^{13r+4}=...700374670114816  --> remainder upon division by 10 is 6;
...

Carcass · Answer

yes true.

The only viable option is to set r=1 in the stem. However, the question as is written in the book is correct.

Thank you

chetan2u · Answer

carcass , I have edited the question and now the answer would be E. So, it’s ok now.

Carcass · Answer

The source is careerlab. Usually the questions, also on the verbal front, are pretty decent.

First time I see those incongruences

BrushMyQuant · Answer

Given that r is a positive integer and we need to find what is the remainder when  138^{13+4r}  is divided by 10

Theory: Remainder of a number by 10 is same as the unit's digit of the number

=> Remainder of  138^{13+4r}  by 10 is same as remainder of  8^{13+4r}  by 10 = unit's digit of  8^{13+4r}

To find the unit's digit of  8^{13+4r}  we need to find the cycle of unit's digit of power of 8

Unit's digit of  8^1  = 8
Unit's digit of  8^2  = 4
Unit's digit of  8^3  = 2
Unit's digit of  8^4  = 6
Unit's digit of  8^5  = 8

So, unit's digit of power of 8 has a cycle of 4.

So, we need to divide the power of 8 by 4 and check the remainder
13 + 4r divided by 4 will give 1 remainder

=> Unit's digit of  8^{13+4r}  = Unit's digit of  8^1  = 8

=> Remainder of  138^{13+4r}  by 10 = 8

So,  Answer will be E 
Hope it helps!

Watch the following video to learn the Basics of Remainders

https://www.youtube.com/watch?v=P0S6j2uTaj4

If r is a positive integer, then what is the remainder when 138

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If r is a positive integer, then what is the remainder when 138

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