You are correct that we can translate statement 1 to: $c(a-1)=0$. However, the conclusion is that either $c=0$ or $a=1$, the word “or” is important. This means that if $a=1$, then statement 1 will hold true, however then $c$ could take on any value. For example, $c$ could be $1$, $2$, or for that any number. This means we do not have a unique value for $c$, and therefore statement 1 alone is insufficient.

Sudhanshu Singh says

why can not we do this a*c=c

which implies a*c-c=0

which implies c(a-1)=0 which in turn c=0 and a=1 what is wrong in this approach

GMAT Quantum says

Hi Sudhanshu,

You are correct that we can translate statement 1 to: $c(a-1)=0$. However, the conclusion is that either $c=0$ or $a=1$, the word “or” is important. This means that if $a=1$, then statement 1 will hold true, however then $c$ could take on any value. For example, $c$ could be $1$, $2$, or for that any number. This means we do not have a unique value for $c$, and therefore statement 1 alone is insufficient.

Dabral