5.5 t ，江t·tll Answer Explanations(2) Given that the least common multiple
(LCM) of Mand Nis 36 and 6 < M < N,
then it is possible that M = (4)(3) = 12 and
N = ( 9 )(^2 ) =^1 8. However,^ it^ is^ also^ possiblethat M = ( 4 )( 3 ) = 12 and N = ( 9 )( 4 ) = (^36) ;
NOT sufficient.
Taking (1) and (2) together, it follows that 6 is a
divisor of Mand Mis a divisor of 36. Therefore,
Mis among the numbers 6, 12, 18, and 36. For
the same reason, N is among the numbers 6,
12, 18, and 36. Since 6 < M < N, it follows that
M cannot be 6 or 36 and N cannot be 6. Thus,
there are three choices for Mand N such that M
< N These three choices are displayed in the
table below, which indicates why only one of the
choices, namely M = 12 and N = 18, satisfies
both (1) and ( 2 ).
／卢
M N GCD LCM
12 18 6 36
(^12 36 12 36)
＼^18 36 18 36
Th e correct answer 1s C·
both statements together are sufficient.
0S07575
- Stations X and Y are connected by two separate,
straight, parallel rail lines that are 250 miles long. Train
P and train Q simultaneously left Station X and Station
Y, respectively, and each train traveled to the other's
point of departure. The two trains passed each other
after traveling for 2 hours. When the two trains
passed, which train was nearer to its destination?
(1) At the time when the two trains passed, train P
had averaged a speed of 70 miles per hour.
(2) Train Q averaged a speed of 55 miles per hour
for the entire trip.Arithmetic
(1) This indicates that Train P had traveled
2(70) = 140 皿les when it passed Trai心It
醮ows that Train P was 250-140 = 110 miles
from its destination and Train Q严S
140 miles from its destination, which means
that Train P was nearer to its destination
when the trains passed each other;
SUFFICIENT.(2) This indicates that Train Qaveraged a speed
of 55 miles per hour for the entire trip, but no
information is given about如speed of Train
P. IfTrain心raveled for 2 hours at an average
speed of 55 miles per hour and Train P
traveled for 2 hours at an average speed of
70 miles per hour, then Train P was nearer to
its destination when the trains passed.
However, if Train心raveled for
2 hours at an average speed of 65 miles
per hour and Train P traveled for 2 hours at
an average speed of 60 miles per hour, then
Train Qwas nearer to its destination when
the trains passed. Note that ifTrain Qtraveled at (120)(55) = 4 7 - miles^1 per hour
140 7
for the remainder of the trip, then its average
speed for the whole trip was 55 miles per
hour; NOT sufficient.
1h e correct answer 1s A·，
statement 1 alone is sufficient.
yxDS01613- In the xy-plane shown, the shaded region consists of
all points that lie above the graph of y = x^2 - 4x and
below the x-axis. Does the point (a,b) (not shown) lie in
the shaded region if b < 0?
(1) 0<a< 4(2) a (^2) - 4a < b
Algebra
In order for (a,b) to lie in the shaded region,
it must lie above the graph of y = x 2 - 4x and
below the x-axis. Since b < 0, the point (a,b) lies
below the x-axis. In order for (a,b) to lie above
the graph of y =正- 4x, it must be true that b >
a^2 - 4a.
( 1 ) This indicates that O <a< 4. If a= 2, then
a^2 - 4a = (^22) - 4 ( (^2) ) = - 4, so if b = -1, then
b > a^2 - 4a and (a,b) is in the shaded
region. But if b = -5, then b < a^2 - 4a and