13. We can check a multiplication $c=a b$ by determining whether the congruence $c \equiv a b(\bmod m)$ is valid, where $m$ is any modulus. If we find that $c \neq a b(\bmod m)$, then we know an error has been made. When we take $m=9$ and use the fact that an integer in decimal notation is congruent modulo 9 to the sum of its digits, this check is called casting out nines. Check each of the following multiplications by casting out nines
a) $875961 \cdot 2753=2410520633$
b) $14789 \cdot 23567=348532367$
c) $24789 \cdot 43717=1092700713$
d) Are your checks foolproof?
13. a) incorrect b) incorrect c) passes casting out nines check
d) no, for example part (c) is incorrect, but passes check