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Thread: Find Divisor

  1. #1
    Super Member harpazo's Avatar
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    Find Divisor

    A number when divided by a divisor leaves a remainder of 16. When twice the original number is divided by the same divisor, the remainder is 7. What would the divisor be?

    My set up:

    Let n = original number

    Let d = divisor

    n/d = 16

    2n/d = 7

    We have two equations in 2 unknowns.

    Correct?
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  2. #2
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    Re: Find Divisor

    when $\displaystyle n $ is divided by $\displaystyle d $ and leaves a remainder of $\displaystyle r$

    the equation for this is

    $\displaystyle n=dq+r$

    where $\displaystyle q $ is the quotient

    so your first equation is of the form

    $\displaystyle n=dq+16$
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  3. #3
    Super Member harpazo's Avatar
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    Re: Find Divisor

    Quote Originally Posted by Idea View Post
    when $\displaystyle n $ is divided by $\displaystyle d $ and leaves a remainder of $\displaystyle r$

    the equation for this is

    $\displaystyle n=dq+r$

    where $\displaystyle q $ is the quotient

    so your first equation is of the form

    $\displaystyle n=dq+16$
    Is the second equation 2n = dq + 7?
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  4. #4
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    Re: Find Divisor

    the second equation

    $2n=d s+7$

    the two quotients are not identical
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  5. #5
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    Re: Find Divisor

    \begin{align}
    n &= 16 \pmod{d} \\
    \implies 2n &= 32 \pmod{d} &(1)\\[8pt]
    \text{also} \quad 2n &= 7 \pmod{d} &(2) \\[12pt]
    (1) - (2) \quad \implies \quad 0 &= 25 \pmod{d} \\
    \implies \quad d &= 25
    \end{align}
    (Actually it implies that $d$ divides 25, but since we know - or assume - that $16 < d < 32$ we get the result).
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  6. #6
    Super Member harpazo's Avatar
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    Re: Find Divisor

    Thank you everyone.
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  8. #8
    Super Member harpazo's Avatar
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    Re: Find Divisor

    Very good.
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