What are the integer value(S) for x where $$\frac{2x+5}{3x+4}$$ is not in lowest terms?
Any help is appreciated!
When is $\frac{(2x+5)}{(3x+4)}$ not in lowest terms?
1
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elementary-number-theory
divisibility
greatest-common-divisor
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0The will occur exactly at those integers $x$ for which $\gcd(2x+5, 3x+4) \neq 1$. – 2017-02-20
1 Answers
5
$gcd(2x+5,3x+4)=gcd(2x+5,x-1)=gcd(7x,x-1)$, but $gcd(7x,x-1)=7$ if $x=7k+1$ and $gcd(7x,x-1)=1$ in any other case.
Thus the fraction is not in lowest terms only when $x=7k+1$,for any integer k.
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0Hi, could you please explain how gcd(2x+5,3x+4)=gcd(2x+5,x−1)? I'm new to this topic... – 2017-02-21
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0It is a property of gcd.In more detail it holds that: $gcd(a,b)=gcd(a,b-ka)$, where k is an integer. In our example we have that: $gcd(2x+5,3x+4)=gcd(2x+5,3x+4-2x-5)$ – 2017-02-21