DSQ: Graphing complex numbers - GMAT Quantitative
Card 1 of 96
Let 
be a positive integer.
True or false: 
Statement 1: 
is a prime number.
Statement 2: 
is a two-digit number ending in a 7.
Let
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
If 
is a positive integer, then 
if and only if 
is a multiple of 4.
It follows that if 
, 
cannot be a prime number. Also, every multiple of 4 is even, so as an even number, 
cannot end in 7. Contrapositively, if Statement 1 is true and 
is prime, or if Statement 2 is true and if 
ends in 7, it follows that 
is not a multiple of 4, and 
.
If
It follows that if
← Didn't Know|Knew It →
Let 
and 
be real numbers.
What is the product of 
and its complex conjugate?
Statement 1: 
Statement 2: 
Let
What is the product of
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number 
is 
, and
.
Therefore, Statement 1 alone, which gives that 
, provides sufficient information to answer the question, whereas Statement 2 provides unhelpful information.
The complex conjugate of an imaginary number
Therefore, Statement 1 alone, which gives that
← Didn't Know|Knew It →
Let 
and 
be real numbers.
What is the product of 
and its complex conjugate?
Statement 1: 
Statement 2: 
Let
What is the product of
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number 
is 
, and
.
Therefore, it is necessary and sufficient to know the values of both 
and 
in order to answer the problem. Each statement alone gives only one of these values, so each statement alone provides insufficient information; the two together give both, so the two statements together provide sufficient information.
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know the values of both
← Didn't Know|Knew It →
Let 
be a positive integer.
Evaluate 
.
Statement 1: 
is a multiple of 16.
Statement 2: 
is a multiple of 20.
Let
Evaluate
Statement 1:
Statement 2:
Tap to reveal answer
The value of 
, 
a positive integer, is equal to 
, where 
is the remainder of the division of 
by 4. Either statement alone is enough to prove that 
is divisible by 4, since, if a number is divisible by a given number (16 or 20 in these statements), it is divisible by any factor of that number (with 4 being a factor of both).
The value of
← Didn't Know|Knew It →
Let 
be a positive integer.
Evaluate 
.
Statement 1: 
is a multiple of 3.
Statement 2: 
is a multiple of 7.
Let
Evaluate
Statement 1:
Statement 2:
Tap to reveal answer
Assume that both statements are true. The value of 
, 
a positive integer, is equal to 
, where 
is the remainder of the division of 
by 4, so we can use this fact to show that insufficient information is provided.
Case 1: 
.
, so
Case 2: 
.
, so
In both cases, both statements are true, but the value of 
differs.
Assume that both statements are true. The value of
Case 1:
Case 2:
In both cases, both statements are true, but the value of
← Didn't Know|Knew It →
Let 
and 
be real numbers.
What is the product of 
and its complex conjugate?
Statement 1: 
Statement 2: 
Let
What is the product of
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number 
is 
, and
.
Statement 1 alone provides insufficient information, as seen in these two scenarios, both of which feature values of 
and 
that add up to 12:
Case 1: 
Then 
, and the product of this number and its complex conjugate is 
.
Case 2: 
Then 
, and the product of this number and its complex conjugate is 
.
The two cases result in different products.
For a similar reason, Statement 2 alone provides insufficient information.
If both statements are assumed to be true, they form a system of equations that can be solved as follows:
Backsolve:
Since we know that 
and 
, then we know that the desired product is 
.
The complex conjugate of an imaginary number
Statement 1 alone provides insufficient information, as seen in these two scenarios, both of which feature values of
Case 1:
Then
Case 2:
Then
The two cases result in different products.
For a similar reason, Statement 2 alone provides insufficient information.
If both statements are assumed to be true, they form a system of equations that can be solved as follows:
Backsolve:
Since we know that
← Didn't Know|Knew It →
Let 
and 
be real numbers.
From the number 
, subtract its complex conjugate. What is the result?
Statement 1: 
Statement 2: 
Let
From the number
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number 
is 
, and
.
Therefore, it is necessary and sufficient to know 
in order to answer the question. Neither statement alone gives this information. However, the first statement can be rewritten by factoring out 
as a difference of squares:
Since 
, then by substitution,
A system of linear equations has now been formed; subtract both sides of the equations as follows:
We need go no further; since 
, the desired difference is 
.
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know
Since
A system of linear equations has now been formed; subtract both sides of the equations as follows:
We need go no further; since
← Didn't Know|Knew It →
Let 
and 
be real numbers.
From the number 
, subtract its complex conjugate. What is the result?
Statement 1: 
Statement 2: 
Let
From the number
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number 
is 
, and
.
Therefore, it is necessary and sufficient to know 
in order to answer the question. Neither statement alone gives this information. However, the two statements together form a linear system that can be solved as follows:
We need go no further; since 
, the desired sum is 
.
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know
We need go no further; since
← Didn't Know|Knew It →
Let 
and 
be real numbers.
What is the sum of 
and its complex conjugate?
Statement 1: 
Statement 2: 
Let
What is the sum of
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number 
is 
, and
.
Therefore, it is necessary and sufficient to know 
in order to answer the question. Neither statement alone gives this information. However, the two statements together form a linear system that can be solved as follows:
We need go no further; since 
, this is the desired sum.
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know
We need go no further; since
← Didn't Know|Knew It →
Let 
and 
be real numbers.
What is the sum of 
and its complex conjugate?
Statement 1: 
Statement 2: 
Let
What is the sum of
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number 
is 
, and
.
We show, however, that the two statements are insufficient to determine the sum by examining two scenarios.
Case 1: 
.
, and since 
, 
. The conditions of both statements are satisfied.
The sum of the numbers is 
.
Case 2: 
.
, and since 
, 
. The conditions of both statements are satisfied.
The sum of the numbers is 
.
In both cases, the conditions of both statements are satisfied, but the sum of the number and its complex conjugate differs between the two.
The complex conjugate of an imaginary number
We show, however, that the two statements are insufficient to determine the sum by examining two scenarios.
Case 1:
The sum of the numbers is
Case 2:
The sum of the numbers is
In both cases, the conditions of both statements are satisfied, but the sum of the number and its complex conjugate differs between the two.
← Didn't Know|Knew It →
Let 
and 
be real numbers.
From the number 
, subtract its complex conjugate. What is the result?
Statement 1: 
Statement 2: 
Let
From the number
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number is , and
.
Therefore, it is necessary and sufficient to know in order to answer the question. Statement 1 does not give this value, and is unhelpful here; Statement 2 does give this value.
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know
← Didn't Know|Knew It →
Let 
and 
be real numbers.
What is the sum of 
and its complex conjugate?
Statement 1: 
Statement 2: 
Let
What is the sum of
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number 
is 
, and the sum of the two numbers is
.
Therefore, it is necessary and sufficient to know 
in order to answer the question. Statement 1 alone gives this information; Statement 2 does not, and it is unhelpful.
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know
← Didn't Know|Knew It →
Let be a positive integer.
True or false:
Statement 1: is a prime number.
Statement 2: is a two-digit number ending in a 7.
Let
True or false:
Statement 1:
Statement 2:
Tap to reveal answer
If is a positive integer, then if and only if is a multiple of 4.
It follows that if , cannot be a prime number. Also, every multiple of 4 is even, so as an even number, cannot end in 7. Contrapositively, if Statement 1 is true and is prime, or if Statement 2 is true and if ends in 7, it follows that is not a multiple of 4, and .
If
It follows that if
← Didn't Know|Knew It →
Let and be real numbers.
What is the product of and its complex conjugate?
Statement 1:
Statement 2:
Let
What is the product of
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number is , and
.
Therefore, Statement 1 alone, which gives that , provides sufficient information to answer the question, whereas Statement 2 provides unhelpful information.
The complex conjugate of an imaginary number
Therefore, Statement 1 alone, which gives that
← Didn't Know|Knew It →
Let and be real numbers.
What is the product of and its complex conjugate?
Statement 1:
Statement 2:
Let
What is the product of
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number is , and
.
Therefore, it is necessary and sufficient to know the values of both and in order to answer the problem. Each statement alone gives only one of these values, so each statement alone provides insufficient information; the two together give both, so the two statements together provide sufficient information.
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know the values of both
← Didn't Know|Knew It →
Let be a positive integer.
Evaluate .
Statement 1: is a multiple of 16.
Statement 2: is a multiple of 20.
Let
Evaluate
Statement 1:
Statement 2:
Tap to reveal answer
The value of , a positive integer, is equal to , where is the remainder of the division of by 4. Either statement alone is enough to prove that is divisible by 4, since, if a number is divisible by a given number (16 or 20 in these statements), it is divisible by any factor of that number (with 4 being a factor of both).
The value of
← Didn't Know|Knew It →
Let be a positive integer.
Evaluate .
Statement 1: is a multiple of 3.
Statement 2: is a multiple of 7.
Let
Evaluate
Statement 1:
Statement 2:
Tap to reveal answer
Assume that both statements are true. The value of , a positive integer, is equal to , where is the remainder of the division of by 4, so we can use this fact to show that insufficient information is provided.
Case 1: .
, so
Case 2: .
, so
In both cases, both statements are true, but the value of differs.
Assume that both statements are true. The value of
Case 1:
Case 2:
In both cases, both statements are true, but the value of
← Didn't Know|Knew It →
Let and be real numbers.
What is the product of and its complex conjugate?
Statement 1:
Statement 2:
Let
What is the product of
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number is , and
.
Statement 1 alone provides insufficient information, as seen in these two scenarios, both of which feature values of and that add up to 12:
Case 1:
Then , and the product of this number and its complex conjugate is .
Case 2:
Then , and the product of this number and its complex conjugate is .
The two cases result in different products.
For a similar reason, Statement 2 alone provides insufficient information.
If both statements are assumed to be true, they form a system of equations that can be solved as follows:
Backsolve:
Since we know that and , then we know that the desired product is .
The complex conjugate of an imaginary number
Statement 1 alone provides insufficient information, as seen in these two scenarios, both of which feature values of
Case 1:
Then
Case 2:
Then
The two cases result in different products.
For a similar reason, Statement 2 alone provides insufficient information.
If both statements are assumed to be true, they form a system of equations that can be solved as follows:
Backsolve:
Since we know that
← Didn't Know|Knew It →
Let and be real numbers.
From the number , subtract its complex conjugate. What is the result?
Statement 1:
Statement 2:
Let
From the number
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number is , and
.
Therefore, it is necessary and sufficient to know in order to answer the question. Neither statement alone gives this information. However, the first statement can be rewritten by factoring out as a difference of squares:
Since , then by substitution,
A system of linear equations has now been formed; subtract both sides of the equations as follows:
We need go no further; since , the desired difference is .
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know
Since
A system of linear equations has now been formed; subtract both sides of the equations as follows:
We need go no further; since
← Didn't Know|Knew It →
Let and be real numbers.
From the number , subtract its complex conjugate. What is the result?
Statement 1:
Statement 2:
Let
From the number
Statement 1:
Statement 2:
Tap to reveal answer
The complex conjugate of an imaginary number is , and
.
Therefore, it is necessary and sufficient to know in order to answer the question. Neither statement alone gives this information. However, the two statements together form a linear system that can be solved as follows:
We need go no further; since , the desired sum is .
The complex conjugate of an imaginary number
Therefore, it is necessary and sufficient to know
We need go no further; since
← Didn't Know|Knew It →