please tell me how to solve this equation :
x/20 = 7 ( mod 5 )
I tried too many methods but it does not work.
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1Try writing $\frac{x}{20} = 2 + 5\cdot q$. Then, $x=140$, perhaps? – Christopher K Nov 10 '13 at 16:02
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It depends on where you are looking for $x$. Most likely you search for $x\in\Bbb Z$ when you are doing congruence relations; in that case $x$ must be divisible by$~20$ so that $x/20$ will be integer. Now multiply the whole equation by$~20$ to get $x\equiv 140\pmod{100}$ which gives the solution; it can also be written $x\equiv 40\pmod{100}$. Indeed $x=40$ satisfies $x/20=2\equiv7\pmod5$ and also for instance $x=-60$ gives $x/20=-3\equiv7\pmod5$.
Marc van Leeuwen
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thank you, but 40 in mod 5 = 0 so that why i can not solve this probem. is it correct if i said x =40 ?? – reem Nov 10 '13 at 16:10
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The solution is a class of integers modulo$~100$ (namely those congruent to$~40$ modulo$~100$), not a class modulo$~5$. If the question were to solve $x/20=7$ as an equation in $\Bbb Z/5\Bbb Z$, then the question is ill posed, since in $\Bbb Z/5\Bbb Z$ there is no such thing as division by$~20$; this is the motivation of the first sentence of my answer. But the question says "$\equiv\pmod 5$" which is just a relation on$~\Bbb Z$ (or possibly on some larger set; it is quite normal to say $\pmod{2\pi}$ for elements of $\Bbb C$). Note that $2x\equiv 6\pmod{10}$ is solved by $x\equiv3\pmod5$ too. – Marc van Leeuwen Nov 10 '13 at 17:26
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In mod 5, 20=0, so you are trying to divide by zero. It won't work.
Empy2
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The equation is in mod5 arithmetic. Why is only the equals sign mod5, and not the divide sign? – Empy2 Nov 10 '13 at 16:23