How to multiply even numbers - GRE Quantitative Reasoning
Card 0 of 6
Which of the following integers has an even integer value for all positive integers 
and 
?
Which of the following integers has an even integer value for all positive integers
There are certain patterns that can be used to predict whether the product or sum of numbers will be odd or even. The sum of two odd numbers is always even, as is the sum of two even numbers. The sum of an odd number and an even number is always odd. In multiplication the product of two odd numbers is always odd. While the product of even numbers, as well as the product of odd numbers multiplied by even numbers is always even. So for this problem we need to find scenarios where the only possibile answers are even. 
can only result in even numbers no matter what positive integers are used for 
and 
, because 
must can only result in even products; the same can be said for 
. The rules provide that the sum of two even numbers is even, so 
is the answer.
There are certain patterns that can be used to predict whether the product or sum of numbers will be odd or even. The sum of two odd numbers is always even, as is the sum of two even numbers. The sum of an odd number and an even number is always odd. In multiplication the product of two odd numbers is always odd. While the product of even numbers, as well as the product of odd numbers multiplied by even numbers is always even. So for this problem we need to find scenarios where the only possibile answers are even.
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If 
is even, and 
is odd. Which of the following must be odd?
If
To solve, pick numbers to represent 
and 
. Let 
and 
. Now try each of the equations given:
.
Only 
works and is thus our answer.
To solve, pick numbers to represent
Only
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is even
is even
Quantity A:
The remainder of 
Quantity B:
1
Quantity A:
The remainder of
Quantity B:
1
Begin by considering our options. For 
, it is either the case that:
is even and 
is odd, or,
is even and 
is even, or,
is odd and 
is even
Now, for 
, it is either the case that:
and 
are even, or,
and 
are odd
Now, for 
, it cannot be the case that both are odd. This means that the only viable option for 
is the case when both are even.
Therefore, 
must be even, meaning that the remainder of 
must be 
.
Therefore, quantity B is larger.
Begin by considering our options. For
Now, for
Now, for
Therefore,
Therefore, quantity B is larger.
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is even
is even
Therefore, which of the following must be true about 
?
Therefore, which of the following must be true about
Recall that when you multiply by an even number, you get an even product.
Therefore, we know the following from the first statement:
is even or 
is even or both 
and 
are even.
For the second, we know this:
Since 
is even, therefore, 
can be either even or odd. (Regardless of what it is, we can get an even value for 
.)
Based on all this data, we can tell nothing necessarily about 
. If 
is even, then 
is even, even if 
is odd. However, if 
is odd while 
is even, then 
will be even.
Recall that when you multiply by an even number, you get an even product.
Therefore, we know the following from the first statement:
For the second, we know this:
Since
Based on all this data, we can tell nothing necessarily about
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In a group of philosophers, 
are followers of Durandus. Twice that number are followers of Ockham. Four times the number of followers of Ockham are followers of Aquinas. One sixth of the number of followers of Aquinas are followers of Scotus. How many total philosophers are in the group?
In a group of philosophers,
In a group of philosophers, 
are followers of Durandus. Twice that number are followers of Ockham. Four times the number of followers of Ockham are followers of Aquinas. One sixth of the number of followers of Aquinas are followers of Scotus. How many total philosophers are in the group?
To start, let's calculate the total philosophers:
Ockham: 
* , or 
Aquinas: 
* , or 
Scotus: 
divided by 
, or 
Therefore, the total number is:
In a group of philosophers,
To start, let's calculate the total philosophers:
Ockham:
Aquinas:
Scotus:
Therefore, the total number is:
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Assume 
and 
are both even whole numbers.
What is a possible solution for 
?
Assume
What is a possible solution for
When two even numbers are multiplied, they must equal an even number. Also, since both variables are said to be even whole numbers, the answer must fit the requirement that its factors are two even numbers multiplied by one another. The only answer that fits both requirements is 
which can be factored into the even whole numbers 
and 
.
When two even numbers are multiplied, they must equal an even number. Also, since both variables are said to be even whole numbers, the answer must fit the requirement that its factors are two even numbers multiplied by one another. The only answer that fits both requirements is
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