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What is the remainder when |5^{13} -6^{14}|

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What is the remainder when |5^{13} -6^{14}|  [#permalink]

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New post Updated on: 13 Aug 2018, 01:14
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  5% (low)

Question Stats:

86% (01:20) correct 14% (01:52) wrong based on 170 sessions

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What is the remainder when |\(5^{13} -6^{14}\)| is divided by 10, where the symbol | | represents the modulus function.

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

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Originally posted by EgmatQuantExpert on 27 Feb 2018, 08:57.
Last edited by EgmatQuantExpert on 13 Aug 2018, 01:14, edited 2 times in total.
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Re: What is the remainder when |5^{13} -6^{14}|  [#permalink]

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New post 27 Feb 2018, 09:48
EgmatQuantExpert wrote:

Question:



What is the remainder when |\(5^{13} -6^{14}\)| is divided by 10, where the symbol | | represents the modulus function.

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


When a number is divided by 10, then the unit's digit of the number will be the remainder.

\(|5^{13}-6^{14}|=|6^{14}-5^{13}|\)

Now we know that \(5\) raised to any power will have unit's digit of \(5\) and \(6\) raised to any power will have unit's digit \(6\). so \(5^{13}\) will have unit's digit \(5\)

Similarly \(6^{14}\) will have unit's digit \(6\)

so unit's digit of \(|6^{14}-5^{13}|=|6-5|=1\)

Hence the remainder will be \(1\)

Option B
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Re: What is the remainder when |5^{13} -6^{14}|  [#permalink]

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New post 27 Feb 2018, 22:30

Solution:



We are given an expression and asked to find out the remainder when the expression is divided by \(10\).
    • Per our conceptual knowledge, if a number is divided by \(10\), the remainder is equal to the units digit of the number itself.
    Thus, this question is asking us to find the units of the expression.
    Now, let us find the units digit of both the numbers.
    • \(5^{13}\)
        o \(5\) has a cyclicity of \(1\)
        o So, the units digit in \(5^{13} = 5\)
    • \(6^{14}\)
        o \(6\) has a cyclicity of \(1\)
        o So, the units digit in \(6^{14} = 6\)
    Now, between \(5^{13}\) and \(6^{14}, 6^{14}\) is larger and the value of \(5^{13}-6^{14}\) will be negative. However, the expression \(5^{13} - 6^{14}\) is between a modulus sign, thus, the final result will be positive.
    This means:
    • \(|5^{13} – 6^{14}|\) =\(|5 – 6|\) units digit = \(|-1|\)units digit = \(1\)
    • Thus, the units digit of \(|5^{13} - 6^{14}| = 1\)
Thus,\(|5^{13} -6^{14}|\) when divided by \(10\) will leave a remainder of \(1\), and hence the correct answer is Option B.
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Re: What is the remainder when |5^{13} -6^{14}| &nbs [#permalink] 27 Feb 2018, 22:30
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