Re: Is n/18 an integer? (1) 5n/18 is an integer. (2) 3n/18 is an integer.
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05 Sep 2020, 13:05
(1) INSUFFICIENT: We are told that 5n/18 is an integer. This does not allow us to determine whether n/18
is an integer. We can come up with one example where 5n/18 is an integer and where n/18 is NOT an integer.
We can come up with another example where 5n/18 is an integer and where n/18 IS an integer.
Let's first look at an example where 5n/18 is equal to the integer 1.
If 5n/18 = 1 then n/18 = 1/5 - In this case n/18 is NOT an integer.
Let's next look at an example where 5n/18 is equal to the integer 15.
If 5n/18 = 15, then n/18 = 3 - In this case n/18 IS an integer.
Thus, Statement (1) is NOT sufficient.
(2) INSUFFICIENT: We can use the same reasoning for Statement (2) that we did for statement (1). If 3n/18
is equal to the integer 1, then n/18 is NOT an integer. If 3n/18 is equal to the integer 9, then n/18 IS an integer.
(1) AND (2) SUFFICIENT: If 5n/18 and 3n/18 are both integers, n/18 must itself be an integer. Let's test some
examples to see why this is the case.
The first possible value of n is 18, since this is the first value of n that ensures that both 5n/18 and 3n/18 are
integers. If n = 18, then n/18 is an integer. Another possible value of n is 36. (This value also ensures that both
5n/18 and 3n/18 are integers). If n = 36, then n/18 is an integer.
A pattern begins to emerge: the fact that 5n/18 AND 3n/18 are both integers limits the possible values of n to
multiples of 18. Since n must be a multiple of 18, we know that n/18 must be an integer. The correct answer is
C.
Another way to understand this solution is to note that according to (1), n = (18/5)*integer, and according to
(2), n = 6*integer. In other words, n is a multiple of both 18/5 and 6. The least common multiple of these two
numbers is 18. In order to see this, write 6 = 30/5. The LCM of the numerators 18 and 30 is 90. Therefore, the
LCM of the fractions is 90/5 = 18.
The correct option is C.