Interchanging Divisors And Remainders

Number Theory Level 2

True or False?

\quad If x x is an integer satisfying 2 x ≡ 5 m o d 7 2x \equiv 5 \bmod 7 , then 5 x ≡ 2 m o d 7 5x \equiv 2 \bmod 7 .

True False

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

Jesse Nieminen
Jul 20, 2016

Relevant wiki: Modular Arithmetic - Multiplication

2 x m o d 7 = 5 ⇔ 2 x ≡ 5 ( m o d 7 ) ⇔ 10 x ≡ 25 ≡ 4 ( m o d 7 ) ⇔ 5 x ≡ 2 ( m o d 7 ) ⇔ 5 x m o d 7 = 2 \begin{aligned} 2x \mod{7} = 5 &\Leftrightarrow 2x \equiv 5 \pmod{7} \\ &\Leftrightarrow 10x \equiv 25 \equiv 4 \pmod{7} \\ &\Leftrightarrow 5x \equiv 2 \pmod{7} \\ &\Leftrightarrow 5x \mod 7 = 2 \end{aligned}

Hence, the answer is True \boxed{\text{True}} .

4 Helpful 0 Interesting 0 Brilliant 1 Confused

A slightly better approach is to multiply the first equation by 6 directly, which is the combination of "multiply by 5 and divide by 2".

What is the reason behind the interchange of coefficients? IE How can we determine what sets of numbers ( a , b , c ) (a, b, c ) satisfy ( a x ≡ b ( m o d c ) ⇔ b x ≡ a ( m o d c ) (ax \equiv b \pmod{c} \Leftrightarrow bx \equiv a \pmod{c} ?

Calvin Lin Staff - 4 years, 10 months ago

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