How to find $\frac{146}7 \mod{7}$?

I understand that if $$\gcd{(b,c)}=1$$ then we can find $$\frac{a}b\mod{c}$$ by writing $$x\equiv \frac{a}b\mod{c}$$ $$bx\equiv a\mod{c}$$ then reducing $$a$$ and solving the modular equation by finding the multiplicative inverse of $$b\mod{c}$$. But when $$\gcd{(b,c)}\ne1$$, in particular if $$b=c$$ then how can one find a value for $$\frac{a}b\mod{c}$$ For example, the number $$\frac{146}7\equiv\frac{48}7\mod{7}$$ according to Wolfram: Alpha. How is this defined?

• Well, $7x\equiv146\pmod7$ has no integer solutions.... – Lord Shark the Unknown Apr 7 at 17:09
• You don't. $7\equiv 0$ and you can't divide by zero. Ever. – Oscar Lanzi Apr 7 at 17:09
• $146 \equiv 48\pmod{7}$, so they probably just reduced the numerator. – Arturo Magidin Apr 7 at 17:30
• It looks like they just repeatedly subtract $7$ from the value so that it falls within the range $0\lt x\lt7$ – Peter Foreman Apr 7 at 17:31

So in the case of $$\dfrac{146}{7}$$ mod 7, wolfram alpha reduced the numerator by multiplying the mod by the denominator to get 146 mod 49. The reason being for you able to do this is $$\dfrac{146}{7}$$ mod 7 can be written as $$\dfrac{146}{7}$$ $$\pm$$ $$\dfrac{7x}{1}$$ which then can be written as $$\dfrac{146}{7}$$ $$\pm$$ $$\dfrac{49x}{7}$$. From here on, you can just mod down the top number to get $$\dfrac{48}{7}$$ mod 7 as the smallest positive solution.
By abusing notation on can define the modulus $$x \pmod n$$ by looking for the real number $$y$$ in the range $$[0,n)$$ so that the difference $$x-y$$ is an integer multiple of $$n$$. In this case exactly this happens: $$\frac{146}{7}\pmod 7=\frac{140}{7}+\frac67\pmod 7=20+\frac67 \pmod 7=6+\frac67\pmod 7$$
$$\frac{158}{7}\pmod{9}=\frac{154}{7}+\frac{4}{7}\pmod 9=22+\frac{4}{7}\pmod 9=4+\frac{4}{7}\pmod 9.$$
The real function that defines such operation ($$\pmod n$$'') is thus $$f(x)=x-\left(\lfloor{\frac{x}n\rfloor}\cdot n\right)$$