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GMAT/Data Sufficiency - How Many m and n's?
How Many m and n's?
What is the positive integer n?
(1) For every positive integer m, the product m(m + 1)(m + 2) ... (m + n) is divisible by 16
(2) n2 - 9n + 20 = 0
(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.
ОТВЕТЫ И КОММЕНТАРИИ
Решение
(1) INSUFFICIENT: For a number to be divisible by 16, it must be divisible by 2 four times. The expression m(m + 1)(m + 2) ... (m + n) represents the product of n + 1 consecutive integers. For example if n = 5, the expression represents the product of 6 consecutive integers.
To find values of n for which the product will be divisible by 16 let's first consider n = 3. This implies a product of four consecutive integers. In any set of four consecutive integers, two of the integers will be even and two will be odd. In addition one of the two even integers will be a multiple of 4, since every other even integer on the number line is a multiple of 4 (4 yes, 6 no, 8 yes, etc...). This accounts for a total of three 2's that can definitely divide into the product of 4 consecutive integers. This however is not enough.
With five integers (i.e. n = 4), there may or may not be an additional 2: there could be two or three even integers. Six integers (i.e. n = 5) will guarantee three even integers and a product that is divisible by four 2's. Anything above six integers will naturally be divisible by 16 as well, so we can conclude that n is greatern than or equal to 5.
(2) INSUFFICIENT: This expression factors to (n - 4)(n - 5) = 0. There are two solutions for n, 4 or 5.
(1) AND (2) TOGETHER SUFFICIENT: If n must be greater than or equal to 5 and either 4 or 5, then n must be equal to 5.
Henrik Ibsen's plays posed as great a challenge to middle-class Scandinavians'
expectations of the drama that almost a century later Edward Albee will offer
to theater-goers in America.
A) that almost a century later, Edward Albee will offer
B) that, almost a century later, Edward Albee would offer
C) as, almost a century later, Edward Albee did offer
D) just as, almost a century later, Edward Albee offered
E) as, almost a century later, Edward Albee would offer
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