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Difficulty:
45%
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Question Stats:
59% (01:37) correct
41%
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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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A. 1
B. 2
C. 4
D. 6
E. 8
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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
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
=> \(138^{12r+r+4}\)
=> \(138^{12r+4}*138^r\)
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
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tanyas311
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.
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Power of Tiny Gains
1.01^(365) = 37.78
0.99^(365) = 0.03
Power of Tiny Gains
1.01^(365) = 37.78
0.99^(365) = 0.03
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
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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
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carcass
carcass please edit the question. Thank you!
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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.
_________________
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.
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carcass
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.
_________________
carcass
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;
...
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gmatophobia
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
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carcass, I have edited the question and now the answer would be E. So, it’s ok now.
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The source is careerlab. Usually the questions, also on the verbal front, are pretty decent.
First time I see those incongruences
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First time I see those incongruences
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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
_________________
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
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All Quant Topics Theory || Last Two Digits of Exponents Ending with 1, 2, 3, 4, 5, 6, 7, 8, 9 || Units' Digit Problems || Even and Odd || Divisibility Rules || Positive and Negative || Absolute Value Basics || Inequalities || Absolute Value + Inequality || Functions and Custom Characters || LCM & GCD || Remainders || Factors and Multiples || Arithmetic and Geometric Progression || Co-ordinate Geometry || Circles || Probability: Coin Toss || Work Rate || Statistics || Stem & Leaf Plot || Box And Whisker Plot
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