whew! this one is a serious cacophony of rephrasing and interpretation.
let's translate:
m is not a factor of p.
if m
were a factor of p, then the remainder upon dividing p by m would be 0.
therefore, we can translate the above statement as follows:
"
the remainder upon dividing p by m is not 0."
in other words, it's an integer greater than 0.
--
the question:
is r > 1 ?
here's a
HIGH-LEVEL INTERPRETATION of this problem.
if the remainder WERE 1, then p would be 1 more than a multiple of m.
if this is the case, then p and m CANNOT have any common factors, other than 1. (this is so because all factors of m are factors of (p-1), which is a multiple of m; a number greater than 1 can't be a factor of both (p-1) and p, which are consecutive integers.)
therefore,
if m and p have common factors, then the answer to this question is YES.
(note that the converse is not necessarily true: even if there are
no common factors, the answer still
could be yes. for instance, 17 divided by 6 leaves a remainder of 5, even though 17 and 6 have no common factors. but, if we can establish that there
are common factors, then that's enough to show that the answer is Yes.)
--
statement (1)
if this is true, then m and p have the factor 2 in common, so, YES.
sufficient.
--
statement (2)
this doesn't tell you whether m and p have common factors.
if m = 5 and p = 6, for instance, then r = 1.
if m = 10 and p = 15, then r = 5, which is > 1.
insufficient.
ans (a)
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