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Example Questions
Example Question #233 : Arithmetic
and
are distinct positive integers.
is an odd quantity. Which of the following must be even?
(You may assume all of these are positive quantities.)
(a)
(b)
(c)
None
(b) and (c) only
(a), (b), and (c)
(a) and (c) only
(a) and (b) only
(a), (b), and (c)
For the product of two positive integers to be odd, both of the integers must themelves be odd. Therefore, if is odd, it follows that both
and
are odd as well. We examine each of the quantities keeping this in mind.
(a) Both and
are odd, so
and
are even. Their product, which is
, must be even.
(b) Since is odd, both
and
are even. Their product,
, is even.
(c) Since both and
are odd, their squares
and
are both odd; it follows that
and
are both even, so their product
is even.
The correct response is that all of the three quantites must be even.
Example Question #71 : Properties Of Integers
and
are distinct positive integers.
is an even quantity. Which of the following must be odd?
(You may assume all of these are positive quantities.)
(a)
(b)
(c)
(b) only
(a) and (c) only
(a), (b), and (c)
(a) only
(b) and (c) only
(b) only
is even, so it follows that one of
and
is even; the other can be even or odd. Let us assume that
will always be even; the argument will be similar if
is assumed to always be even.
(a) is even, so
is odd. If
is even, then
is odd, and
, the product of odd factors, is odd. If
is odd, then
is even, and
, having an even factor, is even. Therefore,
can be even or odd.
(b) is even, so
and
are both odd.
is the product of odd factors and must be odd.
(c) is even, so
is even as well, and
is odd. If
is even, then
is even, and
is odd;
, the product of odd factors, is odd. If
is odd, then
is odd, and
is even;
, having an even factor, is even. Therefore,
can be even or odd.
The correct response is that only (b) need be odd.
Example Question #1791 : Gmat Quantitative Reasoning
and
are distinct positive integers.
is an odd quantity. Which of the following must be even?
(You may assume all of these are positive quantities.)
(a)
(b)
(c)
None of these
(b) and (c) only
(a) and (c) only
(a) and (b) only
(a), (b), and (c)
(a) and (c) only
is an odd quantity if and only if one of
and
is even and the other is odd. We can assume without loss of generality that
is the even quantity and
is the odd quantity, since a similar argument holds if
is even.
(a) is odd, so
is even.
, which has an even factor, is even.
(b) is even and
is odd, so
is even, and
and
are both odd.
, the product of odd factors, is odd.
(c) is odd, so
is odd, and
is even.
, which has an even factor, is even.
The correct response is (a) and (c) only.
Example Question #1791 : Gmat Quantitative Reasoning
Which of the following is not an integer?
Which of the following is not an integer?
The definition of an integer is, "All positive or negative whole numbers, including zero."
Therefore, our answer must pi, because pi is not a whole number. All other options fit the defintion of integers.
Example Question #1792 : Gmat Quantitative Reasoning
Which of the following is an integer?
An integer is an positive or negative whole number, including zero. Eliminate all options which are not whole numbers and you are left with 0!
Example Question #1793 : Gmat Quantitative Reasoning
is a positive integer and has an even number of prime factors. What is
?
We know that is a positive integer with an even factors of prime numbers, for example
could be
or
could be
, in other words,
must be a perfect square. The only possible answer is
, the perfect square of
.
Example Question #1 : Discrete Probability
Race cars for a particular race are numbered sequentially from 12 to 115. What is the probability that a car selected at random will have a tens digit of 1?
There are 104 integers from 12 to 115 inclusive. There are 8 integers from 12 to 19 and 6 integers from 110 to 115 for a total of 14 integers with a tens digit of 1. The probability of selecting a car with a tens digit of 1 is =
.
Example Question #1 : Discrete Probability
Three friends play marbles each week. When they combine their marbles, they have 100 in total. 45 of the marbles are new and the rest are old. 30 are red, 20 are green, 25 are yellow, and the rest are white. What is the probability that a randomly chosen marble is new OR yellow?
Prob(new OR yellow) = P(new) + P(yellow) - P(new AND yellow)
Prob(new) =
Prob(yellow) =
Prob(new AND yellow) =
so P(new OR yellow) =
Example Question #1 : Discrete Probability
Flight A is on time for 93% of flights. Flight B is on time for 89% of flights. Flight A and B are both on time 87% of the time. What is the probabiity that at least one flight is on time?
1.82
0.89
0.87
0.93
0.95
0.95
P(Flight A is on time) = 0.93
P(Flight B is on time) = 0.89
P(Flights A and B are on time) = 0.87
Then P(A OR B) = P(A) + P(B) - P(A AND B) = 0.93 + 0.89 – 0.87 = 0.95
Example Question #3 : Discrete Probability
At a school fair, there are 25 water balloons. 10 are yellow, 8 are red, and 7 are green. You try to pop the balloons. Given that you first pop a yellow balloon, what is the probability that the next balloon you hit is also yellow?
At the start, there are 25 balloons and 10 of them are yellow. You hit a yellow balloon. Now there are 9 yellow balloons left out of 24 total balloons, so the probability of hitting a yellow next is
.
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