PS00774- What is the least integer z for which (0.000125)
(0.0025)(0.00000125) x 10^2 is an integer?
(Al 18
(Bl^10
(Cl^0
(D) - 10
(El -18
Arithmetic
Considering each of the three decimal numbers
in parentheses separately, we know that 0.000125
x 10^6 is the integer 125, 0.0025 x 10^4 is the
integer 25, and 0.00000125 x 10^8 is the integer- We thus know that (0.000125) x 1 06 X
(0.0025) X 1 04 X (0.00000125) X 10^8 =
(0.000125)(0.0025)(0.00000125) X 10^6 X 10^4 X
108 = (0.000125)(0.0025)(0.00000125) X
106 + 4+^8 = (0.000125)(0.0025)(0.00000125) X 10^18
is the integer 125 x 25 x 125. We therefore know
that if z = 18, then (0.000125)(0.0025)
(0.00000125) X 10z is an integer.
Now, if the product 125 x 25 X 125 were divisible
by 10, then for at least one integer z less than 18,
(0.000125)(0.0025)(0.00000125) X 10z would be
an integer. However, each of the three numbers
being multiplied in the product 125 x 25 x 125 is
odd (not divisible by 2). We thus know that 125 x
25 x 125 is not divisible by 2 and is therefore
odd. Because only even numbers are divisible by
10, we know that 125 x 25 x 125
is not divisible by 10. We thus know that 18 is
the least integer z such that (0.000125)(0.0025)
(0.00000125) X 10z is an integer.Note that it is not necessary to perform the
multiplication 125 X 25 X 125.Th e correct answer 1s A.4.5 Answer ExplanationsPS08407
100.The average (arithmetic mean) length per film for a
group of 21 films is t minutes. If a film that runs for
66 minutes is removed from the group and replaced
by one that runs for 52 minutes, what is the average
length per film, in minutes, for the new group of films,
in terms of t?(A)
t+—^2
3(B)
t——^2
3
(Cl 21 t+ 14(D) t+-^3
2(E) t-—^3
2Arithmetic
Let S denote the sum of the lengths, in minutes,
of the 21 films in the original group. Since the
average length is t minutes, it follows that—s = t.
21If a 66-minute film is replaced by a 52-minute
film, then the sum of the lengths of the 21 films
in the resulting group is S - 66 + 52 = S - 14.
Therefore, the average length of the resulting
21 fi1.s— (^14) S 14 2
ms 1s =—-—=t-—.
21 21 21 3
Th e correct answer 1s B.
PS0805!
- An open box in the shape of a cube measuring
50 centimeters on each side is constructed from
plywood. If the plywood weighs 1.5 grams per square
centimeter, which of the following is closest to the total
weight, in kilograms, of the plywood used for the box?
(1 kilogram = 1,000 grams)
(Al 2
(Bl 4
(Cl^8
(D) 13
(El 19