Confusion in symbols of a book

Question

I was reading book "" Concrete Mathematics by

Ronald L. Graham AT&T Bell Laboratories, ] Donald E. Knuth Stanford University, Oren Patashnik Center for Communications Research""

and I cam across these statements

[m/n] 1 if m divides n, otherwise 0 ,,,,,, and [m//n] 1 if m exactly divides n, otherwise 0

Please explain the difference between the two since both explain the same thing if n divides completely m then equal to 1 else 0..

Answer 1

You’ve miscopied the notation. They write $m\backslash n$ to mean that there is an integer $k$ such that $n=mk$; this is their version of the more usual notation $m\mid n$, read ‘$m$ divides $n$’. They write $m\backslash\backslash n$ to mean that there is an integer $k$ such that $n=km$ and $\gcd(k,m)=1$: $m$ divides $n$, and $m$ and $\frac{n}m$ are relatively prime.

Answer 2

I am acquainted with $a\mid b$ for $\exists c\colon b=ac$ and, if $a$ is a prime, $a^n\|b$ for $a^n\mid b\land a^{n+1}\nmid b$. A generalization of $\|$ to arbitrary integers would be to define $a\|b$ as $a|b\land \gcd(a,\tfrac ba)=1$.