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GMAT/Data Sufficiency - Divisibility Dilemma
Divisibility Dilemma
If a and b are consecutive positive integers, and ab = 30x is x a non-integer?
(1) a2 is divisible by 21
(2) 35 is a factor of b2
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.
ОТВЕТЫ И КОММЕНТАРИИ
Решение
First, let's rephrase the complex wording of this question into something easier to handle. The question asks whether x is a non-integer which is the reverse of an easier question: Is x an integer?
Surely, if we can answer this question, we can answer the original question.
We can isolate x by rewriting the given equation as follows: .
In order for x to be an integer, ab must be divisible by 30. In order for a number to be divisible by 30, it must have 2, 3, and 5 as prime factors (since 30 = 2 x 3 x 5 ).
Thus, the question becomes: Does ab have 2, 3, and 5 as prime factors?
We know from the question that a and b are consecutive positive integers. Thus, either a or b is an even number, which means that the product ab must be divisible by 2.
Since we know that 2 is a prime factor of ab, the question can be further simplified: Does ab have 3 and 5 as prime factors?
Statement (1) tells us that 21 is a factor of a2 which means that 3 and 7 are prime factors of a2.
We can deduce from this that 3 and 7 must also be factors of a itself. (How? We know that a is a positive integer, which means that a2 is a perfect square. All prime factors of perfect squares come in pairs. Thus, if a2 is divisible by 3, then a2 must be divisible by a pair of 3's, which means that a itself must be divisible by at least one 3. You can test this using a real value for a2.)
Knowing that 3 is a prime factor of a tells us that 3 is a factor of ab but is not sufficient to answer our rephrased question, since we know nothing about whether 5 is a factor of ab.
Statement (2) tells us that 35 is a factor of b2 which means that 5 and 7 are prime factors of b2.
Using the same logic as for the previous statement, we can deduce that 5 and 7 must be factors of b itself. Knowing that 5 is a prime factor of b tells us that 5 is a factor of ab but it is not sufficient to answer our rephrased question, since we know nothing about whether 3 is a factor of ab.
If we combine both statements, we know that ab must be divisible by both 3 and 5, which is sufficient information to answer the original question. The correct answer is C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
A rectangular plot of land is represented on a map. What are the actual
dimensions of the plot of land?
(1) The length of the rectangular figure on the map representing the actual
plot of land is twice as long as the width.
(2) The map is drawn so that each ^ inch on the map represents an actual
distance of 10 feet.
A)if statement I alone is sufficient to answer the question, but statement 2
alone is not sufficient
B)if statement 2 alone is sufficient to answer the question, but statement I
alone is not sufficient
C)if both statements together are needed to answer the question, but neither
statement alone Is sufficient
D)if either statement by itself is sufficient to answer the question
E)if not enough facts are given to answer the question
Вверх 
