and it follows that the total number of machines
working that minute was 2 + 4 + 6 = 12.1h e correct answer 1s C· ，
both statements together are sufficient.
DS08660- If a and b are constants, what is the value of a?
(1) a< b
2 () t—a)(t-b) = t(^2 + t-12, for all values oft.Algebra
(1)(2)Given that a< b, it is not possible to
determine the value of a. For example, a < b
is true when a = l and b = 2 , and a < b is
true when a = 2 and b = 3; NOT sufficient.
By factoring, what is given can be expressedas (t- a)(t- b) = (t + (^4) )(t- 3), so either a =
-4 and b = 3 , or a = 3 and b = -4; NOT
sufficient.
Taking (1) and (^2 ) together, the relation a< bis
satisfied by only one of the two possibilities given
in the discussion of (2) above, namely a= -4 and
b = 3. Therefore, the value of a is -4.
1h e correct answer 1s C;
both statements together are sufficient.
DS04474
- If xis a positive integer, is奴aninteger?
(1) 拉及isan integer.
(2)高isnot an integer.Algebra
(^1 ) It is given that石=n, or 4x = n^2 , for some
positive integer n. Since 4x is the square
of an integer, it follows that in the prime
factorization of 4x, each distinct prime
factor is repeated an even number of times.
Therefore, the same must be true for出e prime
factorization of x, since the prime factorization
of x only differs from the prime factorization
of 4x by two factors of 2, and hence by an even
number of factors of 2; SUFFICIENT.
( 2 ) Given that忘isnot an integer, it is
possible for嘉tobe an in
fer (for example,
x = l) and it is possible for x to not be an
integer (for example, x = 2); NOT sufficient.
1h e correct answer 1s A;
statement 1 alone is sufficient.5.5 I'ill扦ICI七·r Answer ExplanationsDS16456- If p, q, x, y, and z are different positive integers, which
of the five integers is the median?
(l) p + X < Q
(2) Y< zArithmetic
Since there are five different integers, there
are two integers greater and two integers less
than the median, which is the middle number.(1) No information is given about the order of y
and z with respect to the other three
numbers; NOT sufficient.
(2) This statement does not relate y and z to the
other three integers; NOT sufficient.Because (1) and (2) taken together do not relate
p, x, and q toy and z, it is impossible to tell
which is the median. For example, if p = 3, x = 4,
q = 8,y = 9, and z = 10, then the median is 8, but
if p = 3, x = 4, q = 8,y = l, and z = 2, then the
median is 3.Th e correct answer 1s E;
both statements together are still not sufficient.
DS16277- If w + z = 28, what is the value of wz?
(1 ) wand z are positive integers.
(2 ) wand z are consecutive odd integers.Arithmetic
(1) The fact that w and z are both positive
integers does not allow the values of w and
z to be determined because, for example,
if w = 20 and z = 8, then wz = 160, and if
w = 10 and z = 18, then wz = 180; NOT
sufficient.(2) Since w and z are consecutive odd integers
whose sum is 28, it is reasonable to consider
the possibilities for the sum of consecutive
odd integers: ... , (—5) + (-3) = - 8,(- 3 ) + (-1) = - (^4) , (- (^1) ) + (^1) = 0, 1 + 3 = 4, ... ,
9 + 11 = 20, 11 + 13 = 24, 13 + 15 = 28,
15 + 17 = 3 2 , .... From this list it follows
that only one pair of consecutive odd integers
has 28 for its sum, and hence there is exactly
one possible value for wz.
This problem can also be solved algebraically
by letting the consecutive odd integers w and