Explanation
This "distributing evenly" business will boil down to
divisibility. For example, if there are 5 dealerships, we'll be able to
distribute the cars evenly if the number of cars is divisible by 5. On to the
data statements, separately first.
Statement (1) tells us that we could distribute 3a cars. This whole business is ripe for
analysis by cases, because there are only a few possible numbers of
dealerships: the number of dealerships is
. We'll start with
and
, so
. Those numbers all fit
the conditions in the question and Statement (1), so it's a legal case. We can
distribute 24 cars evenly to 4 dealership. Can we distribute a cars evenly? Since
, we could, and the answer is yes. That's Case I.
We'll try to find a case in which the answer is no. Perhaps
and
? That's not a legal case, because
cars can't be
distributed equally among 5 dealerships. We notice a pattern. The statement is
telling us that
But b must be 4,
5, or 6. With the 3 present, is the only way
will be an
integer that b goes into a?
No, wait: b could be 6 and a could possess a
factor of 2. If
, 3a cars
could be divided evenly among the 6 dealerships, but not a cars. We have obtained a valid case with an answer "no," so
Statement (1) is insufficient.
Statement (2) is similar. It says that
This time we draw a different conclusion. Since b must be 4, 5, or 6, and none of those
numbers goes into 7, b must go into a. That means each of b dealerships can receive an equal share
of a cars, and the answer to the
question is definitively "yes." Statement (2) is sufficient.
The correct answer is (B).
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