Suppose that $x$ and $y$ are congruent modulo $24$, that is, $x \equiv y\ (mod\ 24)$.
Which of the following is not guaranteed to be true?
a. $x$ and $y$ have the same last digit in binary notation
b. $x$ and $y$ have the same last digit in decimal notation
c. $(x \mod 3) = (y \mod 3)$
d. Both $x ≡ y (mod\ 6)$ and $x ≡ y (mod\ 8)$ are true.
From just trying to understand the problem I concluded that (b) and (c) are guaranteed to be true. But what steps can I take to logically answer this question?
$\endgroup$ 11 Answer
$\begingroup$HINT(b) consider 24 and 48.
Note that 2,3,6 and 8 are divisors of 24, while 10 is not.
$\endgroup$ 2
