All practice questions

Question 141Question →

A municipal health board announced that implementing a comprehensive vector-control program is essential for eliminating endemic dengue transmission in urban districts. Last year, the city of Solaria fully funded and executed a comprehensive vector-control program across all its urban districts. Therefore, city health officials concluded that endemic dengue transmission has now been successfully eliminated in Solaria.

Which of the following best describes the reasoning vulnerability in the health officials' argument?

It presumes that a vector-control program cannot be successfully executed without full municipal funding.

It infers that because the vector-control program preceded a reduction in virus transmission, the program must have been the sole cause of that reduction.

It treats a condition that is required for eliminating dengue transmission as though it is sufficient to guarantee that elimination.

It assumes without justification that Solaria's vector-control protocols will remain effective under changing climatic conditions.

It fails to consider that implementing vector-control programs might inadvertently increase human exposure to other pest-borne pathogens.

Show answer & explanation

Answer: It treats a condition that is required for eliminating dengue transmission as though it is sufficient to guarantee that elimination.

Answer

The argument treats a condition that is necessary for eliminating dengue transmission as though it were sufficient to guarantee that outcome.

The correct answer accurately identifies the core logical flaw: the premise states that a comprehensive vector-control program is 'essential' (necessary) for eliminating dengue transmission. However, the health officials conclude that merely executing this program guarantees that dengue transmission has been eliminated. This treats a necessary prerequisite as a sufficient condition for success.

Step-by-Step Solution

1

Identify the premises and conclusion

Premise: Vector-control program is essential (necessary) for dengue elimination. Premise: Solaria implemented the program. Conclusion: Dengue transmission has been eliminated.

Deconstructing the argument structure clarifies the underlying conditional claims.

2

Analyze the conditional logic relationship

The word 'essential' indicates that Vector-Control is a necessary condition for Dengue Elimination. Implementing Vector-Control does not automatically guarantee Elimination.

Differentiating between necessary and sufficient conditions reveals whether the premise logically justifies the conclusion.

3

Pinpoint the logical error

The health officials assume that satisfying a necessary condition guarantees the outcome, thereby confusing a necessary condition for a sufficient one.

Satisfying a prerequisite does not ensure success if other necessary conditions remain unfulfilled.

Key Concept

Confusing Necessary and Sufficient Conditions

Estimated Time:2m 0s

Question 142Question →

A regional health authority recently mandated that all public hospitals in the province implement a post-discharge digital health monitoring application for heart failure patients. Over the two years following the mandate, readmission rates for heart failure patients at these hospitals dropped by 22 percent. The health authority concluded that the remote monitoring application was directly responsible for the reduction in patient readmissions.

Which of the following would it be most useful to establish in order to evaluate the health authority's conclusion?

Whether patients using the digital application experienced an increase in overall life expectancy beyond the standard post-discharge follow-up period.

Whether readmission rates for heart failure patients also experienced a comparable decline at hospitals in neighboring districts that did not introduce the remote monitoring technology during the same timeframe.

Whether the initial implementation of the digital monitoring application led to temporary administrative delays for hospital nursing staff.

Whether the digital monitoring application caused patients to become overly reliant on automated alerts rather than contacting emergency services when severe symptoms developed.

Whether medical staff initially opposed the health authority's mandate on the grounds that software cannot replace in-person physician evaluations.

Show answer & explanation

Answer: Whether readmission rates for heart failure patients also experienced a comparable decline at hospitals in neighboring districts that did not introduce the remote monitoring technology during the same timeframe.

Answer

The correct answer is the option inquiring whether readmission rates for heart failure patients also experienced a comparable decline at hospitals in neighboring districts that did not introduce the remote monitoring technology during the same timeframe.

The correct answer establishes a controlled comparison group. Evaluating whether heart failure readmissions dropped similarly at comparable hospitals that lacked the digital application allows us to determine if the application was the true driver of the 22 percent reduction or if an unmentioned regional trend (e.g., changes in standard drug regimens or overall healthcare policies) produced the change.

Step-by-Step Solution

1

Identify the Premise and Conclusion

Premise: After public hospitals introduced a digital monitoring application, heart failure readmission rates fell by 22 percent over two years. Conclusion: The digital application caused the reduction in readmissions.

Understanding the precise causal link asserted by the argument is necessary before evaluating its validity.

2

Identify the Causal Flaw / Evaluation Need

The argument assumes correlation equals causation and ignores potential confounding variables or regional trends.

To evaluate a causal claim based on a single temporal event sequence, one must test whether an unstated third variable or broader trend produced the observed effect.

3

Apply Variance Test to Options

Comparing the target hospitals with a control group of non-adopting hospitals isolates the impact of the application from external regional variables.

If non-adopting hospitals saw the same drop, the application was not the cause. If non-adopting hospitals saw no drop, the application's causal role is strongly supported.

Key Concept

Evaluating Causal Arguments via Control Group Comparison

Question 143Question →

To mitigate severe summer train delays caused by the thermal expansion of steel tracks, a municipal transit authority plans to coat all exposed rail lines with a solar-reflective ceramic finish. In pilot testing conducted on an unused, stationary track segment, the coating reduced peak surface temperatures by 15°C, keeping the metal well below the critical temperature at which heat-induced warping occurs. The authority concludes that implementing this coating systemwide will successfully eliminate heat-related train slowdowns. Which of the following, if true, most seriously undermines the transit authority's conclusion?

Recent chemical enhancements to ceramic coatings have significantly reduced their purchasing cost while increasing overall solar heat reflectivity.

Air conditioning system failures on older passenger cars account for a larger proportion of passenger delay complaints during summer months than track warping does.

The intense friction generated by the steel wheels of active, heavy train traffic rapidly abrades the ceramic coating, stripping it off the rails within days of operational use.

The thermal expansion threshold of steel rails manufactured within the last five years is slightly higher than that of older steel rails currently in service.

Applying the ceramic coating requires maintenance crews to conduct track work during nighttime hours when ambient temperatures are lowest.

Show answer & explanation

Answer: The intense friction generated by the steel wheels of active, heavy train traffic rapidly abrades the ceramic coating, stripping it off the rails within days of operational use.

Answer

The argument is most seriously weakened by the fact that the friction of active train traffic rapidly abrades and strips the ceramic coating off the rails within days.

The correct answer undermines a key assumption of the plan: that the ceramic coating will remain effective under real-world operating conditions as it did on an unused test track. By showing that mechanical friction from heavy train traffic rapidly strips the coating off the rails, this choice demonstrates that the plan cannot achieve its goal of preventing summer track warping.

Step-by-Step Solution

1

Deconstruct the argument core

Premise: In a pilot test on an unused, stationary track, solar-reflective ceramic coating lowered peak metal surface temperatures by 15°C, preventing thermal warping. Conclusion: Systemwide application on operational tracks will eliminate heat-related slowdowns.

Identifying the author's logical leap from a stationary test environment to real-world operating conditions reveals the underlying assumption.

2

Identify the unstated assumption

The author assumes that conditions on active, heavily trafficked rail lines will not interfere with the coating's durability or thermal performance compared to the stationary test line.

Plan-to-goal arguments rely on the implicit premise that pilot conditions accurately forecast operational realities.

3

Evaluate the impact of new evidence

If high-frequency heavy train traffic removes the coating almost immediately through wheel friction, the coating cannot maintain its reflective benefits, completely undermining the plan's expected outcome.

Showing that real-world operational factors negate the key mechanism demonstrated in the pilot directly breaks the link between the premise and conclusion.

Key Concept

Extrapolation from Pilot Conditions to Operational Environments

Question 144Question →

To reduce order fulfillment errors caused by manual sorting, a major online retailer plans to deploy Autonomous Mobile Robots (AMRs) across all its regional distribution centers. Management notes that in pilot tests at a prototype facility, the automated sorting system reduced mispackaged shipments by 80 percent compared to human sorters. Therefore, management concludes that deploying AMRs nationwide will significantly lower overall operational costs associated with shipping errors.

Which of the following, if true, most seriously weakens management's argument?

The specialized technical labor and maintenance fees required to resolve routine AMR sorting malfunctions cost more per incident than the financial losses previously incurred from human mispackaging errors.

Automated mobile robots require less daily operational downtime for battery recharging than human workers require for mandatory rest breaks.

Competing e-commerce platforms have chosen to outsource fulfillment operations to third-party logistics firms rather than purchasing proprietary sorting technology.

Mispackaged shipments account for a smaller overall percentage of customer service complaints than delivery delays caused by severe weather events.

The initial pilot testing was conducted during daytime shifts, whereas regional fulfillment facilities operate continuous 24-hour schedules.

Show answer & explanation

Answer: The specialized technical labor and maintenance fees required to resolve routine AMR sorting malfunctions cost more per incident than the financial losses previously incurred from human mispackaging errors.

Answer

The statement showing that specialized technical maintenance fees for robot sorting errors exceed the losses from human sorting errors.

The conclusion asserts that deploying AMRs will lower overall operational costs associated with shipping errors. The correct answer weakens this argument by demonstrating an unconsidered cost of the new technology: fixing AMR sorting malfunctions requires expensive specialized technicians whose fees exceed the cost of human sorting mistakes. Consequently, even with fewer physical mispackages, the financial cost associated with sorting errors will increase, defeating management's goal.

Step-by-Step Solution

1

Deconstruct the argument structure

Premise: Pilot tests showed an 80% reduction in mispackaged shipments using AMRs compared to human sorters.
Conclusion: Nationwide deployment of AMRs will significantly lower overall operational costs associated with shipping errors.

Identifying the explicit premise and conclusion isolates the central claim and logical gap.

2

Identify the unstated assumption

The argument assumes that the financial savings achieved by preventing human mispackaging errors will not be offset or exceeded by new costs created by the AMR system itself.

A cost-reduction conclusion requires that total expenditures associated with the new process are net lower than under the old process.

3

Evaluate answer choices for an alternative factor that invalidates the net cost reduction

The option pointing out that specialized technical labor and maintenance fees to fix AMR malfunctions cost more per incident than human sorting errors directly shatters the net savings premise.

Showing that new error-handling costs surpass old error losses proves the total operational costs associated with shipping errors will actually rise.

Key Concept

Weakening Cost-Benefit Arguments

Question 145Question →

For any integer $n$ , which of the following expressions MUST be an even integer?

(n + 1)^3 - (n - 1)^3

n^3 - n + 1

(n + 1)^2 + n^2

3n^2 + 3n + 1

n^2 + n - 3

Show answer & explanation

Answer: $(n + 1)^3 - (n - 1)^3$

Answer

The expression

(n + 1)^3 - (n - 1)^3

MUST be an even integer for all integer values of

n

.

Simplifying the expression

(n + 1)^3 - (n - 1)^3

yields

6n^2 + 2

, which equals

2(3n^2 + 1)

. Since

3n^2 + 1

is always an integer for any integer

n

, multiplying by 2 guarantees the expression is an even integer for all values of

n

, including zero and negative integers.

Step-by-Step Solution

1

Expand both cubic terms algebraically.

(n + 1)^3 = n^3 + 3n^2 + 3n + 1

and

(n - 1)^3 = n^3 - 3n^2 + 3n - 1

.

Expanding the terms allows combining like terms to simplify the expression.

2

Subtract the expanded expression

(n - 1)^3

from

(n + 1)^3

.

(n^3 + 3n^2 + 3n + 1) - (n^3 - 3n^2 + 3n - 1) = 6n^2 + 2

.

The cubic terms (

n^3

) and linear terms (

3n

) cancel out.

3

Factor out a common factor of 2 from the simplified expression.

6n^2 + 2 = 2(3n^2 + 1)

.

Any integer that can be expressed as

2k

, where

k

is an integer, is by definition even. Since

n

is an integer,

3n^2 + 1

is an integer, making

2(3n^2 + 1)

an even integer for all

n

.

Key Concept

Parity rules for algebraic expressions and consecutive integer products

Alternative Method

Instead of algebraic expansion, test values of

n

with different parities, including

n = 0

(even) and

n = 1

(odd). For

n = 0

,

(0 + 1)^3 - (0 - 1)^3 = 1 - (-1) = 2

(even). For

n = 1

,

(1 + 1)^3 - (1 - 1)^3 = 8 - 0 = 8

(even). Because

n + 1

and

n - 1

always have the same parity, their cubes also share the same parity, so their difference is always even.

Estimated Time:1m 30s

Question 146Question →

If $a$ and $b$ are integers such that $a^2b + a$ is an odd integer, which of the following expressions must be an even integer?

a + b

a^2 - b

a + b + 1

ab + a

a^2 + b^2

Show answer & explanation

Answer: $a + b + 1$

Answer

The expression

a + b + 1

must be an even integer.

Factoring

a^2b + a

gives

a(ab + 1)

. For a product of two integers to be odd, both factors must be odd. Therefore,

a

is odd and

ab + 1

is odd. If

ab + 1

is odd, then

ab

must be even. Since

a

is odd,

b

must be even. Evaluating the expression

a + b + 1

:

a

(odd) +

b

(even) + 1 (odd) equals an even integer.

Step-by-Step Solution

1

Factor the given algebraic expression to analyze its parity components.

a^2b + a = a(ab + 1)

.

Factoring allows us to analyze the parity of individual factors whose product is given as odd.

2

Determine the parity of

a

and the factor

(ab + 1)

.

Both

a

and

(ab + 1)

must be odd integers.

The product of two integers is odd if and only if both integer factors are odd.

3

Determine the parity of

b

using the parities established in Step 2.

Since

ab + 1

is odd,

ab

must be even. Because

a

is odd,

b

must be even (including zero).

An odd number multiplied by an even number yields an even number, and adding 1 converts it to an odd number.

4

Evaluate the parity of

a + b + 1

.

\text{odd} + \text{even} + 1 = \text{odd} + 1 = \text{even}

.

Adding two odd numbers (

a

and 1) together with an even number (

b

) results in an even integer.

Key Concept

Parity rules for integer addition and multiplication: Odd × Odd = Odd, Odd × Even = Even, Odd + Even = Odd, Odd + Odd = Even.

Question 147Question →

If $n$ is any integer, which of the following expressions must be an even integer?

3n^2 + 5n + 2

n^2 + n + 1

2n^2 + 3n

n^3 + 2n

n^2 + 3

Show answer & explanation

Answer: $3n^2 + 5n + 2$

Answer

The expression

3n^2 + 5n + 2

must be an even integer for any integer

n

.

The expression

3n^2 + 5n + 2

can be factored as

3n(n+1) + 2

. Since

n

and

n+1

are consecutive integers, one of them must be even, making

n(n+1)

an even integer. Multiplying an even integer by 3 produces an even integer, and adding 2 maintains even parity. Therefore, this expression is guaranteed to be even for all integer values of

n

.

Step-by-Step Solution

1

Analyze the parity property of consecutive integers.

For any integer

n

, one of the terms in the pair

\{n, n+1\}

is even. Hence, the product

n(n+1)

is always an even integer.

The product of an even integer and any integer is always even.

2

Rewrite the target expression

3n^2 + 5n + 2

to isolate the consecutive integer product.

3n^2 + 5n + 2 = 3n^2 + 3n + 2 = 3n(n+1) + 2

.

Algebraic manipulation isolates known parity components.

3

Determine the parity of

3n(n+1) + 2

.

Since

n(n+1)

is even,

3 \times \text{even} = \text{even}

. Then

\text{even} + 2 = \text{even}

.

Multiplying an even integer by an odd integer yields an even integer, and adding an even integer preserves even parity.

Key Concept

Product of consecutive integers

n(n+1)

is always even; zero is an even integer; basic parity rules under addition and multiplication.

Estimated Time:1m 30s

Question 148Question →

If $n$ is an integer such that $1 \le n \le 50$ , how many values of $n$ satisfy the condition that $n^2 + 3n$ is an odd integer?

Show answer & explanation

Answer: 0

Answer

0

For any integer

n

, the expression

n^2 + 3n

can be rewritten as

n(n + 3)

. Since

n

and

n + 3

differ by 3 (an odd number), one factor must be even and the other must be odd. The product of an even integer and an odd integer is always even. Therefore,

n^2 + 3n

is even for all integer values of

n

, meaning there are exactly 0 values of

n

in the given range for which the expression is odd.

Step-by-Step Solution

1

Analyze the algebraic expression

n^2 + 3n

by testing parity cases for any integer

n

.

The parity of the expression depends on whether

n

is even or odd.

Every integer is either even or odd.

2

Evaluate

n^2 + 3n

assuming

n

is even.

n^2

is even and

3n

is even; their sum is even.

Even

\times

Even = Even, and Even

+

Even = Even.

3

Evaluate

n^2 + 3n

assuming

n

is odd.

n^2

is odd and

3n

is odd; their sum is even.

Odd

\times

Odd = Odd, and Odd

+

Odd = Even.

4

Count the number of integers

n

in

1 \le n \le 50

for which

n^2 + 3n

is odd.

Because

n^2 + 3n

is even for all integer values of

n

, zero values of

n

make the expression odd.

No integer

n

satisfies the condition.

Key Concept

Odd and Even Integers (Parity)

Estimated Time:1m 0s

Question 149Question →

For how many positive integers $n$ less than or equal to $120$ is the expression $n^3 - n + 3^n$ an even integer?

Show answer & explanation

Answer: 0

Answer

0

Factoring

n^3 - n

into

(n-1)n(n+1)

shows that this component is the product of three consecutive integers and is therefore always even for any integer

n

. Since 3 is odd,

3^n

is always odd for any positive integer exponent

n

. The sum of an even integer and an odd integer is always odd. Consequently,

n^3 - n + 3^n

is odd for all positive integers

n

, meaning 0 values of

n \le 120

yield an even integer.

Step-by-Step Solution

1

Analyze the algebraic expression by factoring

n^3 - n

.

n^3 - n = n(n^2 - 1) = (n - 1)n(n + 1)

, which represents the product of three consecutive integers.

The product of consecutive integers contains at least one even number, making

n^3 - n

even for all integers

n

.

2

Analyze the exponential term

3^n

.

Since 3 is odd,

3^n

is odd for every positive integer

n

.

An odd integer raised to any positive integer exponent yields an odd integer.

3

Determine the overall parity of the expression

n^3 - n + 3^n

.

\text{even} + \text{odd} = \text{odd}

. The expression is odd for all positive integers

n

.

Adding an odd integer to an even integer results in an odd integer.

4

Count the number of positive integers

n \le 120

for which the expression is even.

There are 0 such integers.

Because the expression is odd for every positive integer

n

, it is never even.

Key Concept

Parity rules of consecutive integer products, odd exponents, and addition of even/odd integers.

Question 150Question →

If $m$ , $n$ , and $p$ are integers such that $m^2 n + n^2 p + p^2 m + m n p$ is an odd integer, which of the following expressions MUST be an even integer?

m n + n p + p m

m + n + p + 1

m n p + 1

m^2 + n^2 + p^2 + 1

m + n + p

Show answer & explanation

Answer: $m + n + p$

Answer

The sum of the three variables,

m + n + p

, MUST be an even integer.

Analyzing the expression

m^2 n + n^2 p + p^2 m + m n p

modulo 2 reveals that

x^2

has the same parity as

x

. Substituting modulo 2 yields

m n + n p + p m + m n p

. Testing all combinations of parity for

m, n, p

shows this expression is odd if and only if exactly two of the three variables are odd and one is even. The sum of two odd integers and one even integer (

m + n + p

) is always even, so

m + n + p

must be an even integer.

Step-by-Step Solution

1

Analyze the parity of the given expression modulo 2.

Since

x^2 \equiv x \pmod 2

for any integer

x

, the given expression

m^2 n + n^2 p + p^2 m + m n p \equiv m n + n p + p m + m n p \pmod 2

.

Squaring an integer does not change its parity.

2

Determine which parity combinations of

m

,

n

, and

p

result in an odd value.

Evaluating

m n + n p + p m + m n p \pmod 2

across all possible parity combinations shows that the expression is odd if and only if exactly two of the variables are odd and exactly one variable is even.

If all three are even or one is odd and two are even, the expression equals 0 (even). If all three are odd,

1 + 1 + 1 + 1 = 4 \equiv 0

(even). Only when exactly two variables are odd (e.g., 1, 1, 0) does

1\cdot 1 + 1\cdot 0 + 0\cdot 1 + 1\cdot 1\cdot 0 = 1

(odd).

3

Test the parity of

m + n + p

under the condition that exactly two variables are odd and one is even.

\text{odd} + \text{odd} + \text{even} = \text{even} + \text{even} = \text{even}

.

Adding two odd integers produces an even integer, and adding an even integer keeps the sum even.

Key Concept

Odd and Even Integers (Parity)

Estimated Time:2m 0s

Question 151Question →

If $p$ and $q$ are integers such that $3p + 2q$ is an odd integer, which of the following statements must be true?

p

must be odd, but

q

can be any integer

p

must be odd, and

q

must be even

Both

p

and

q

must be odd

p

must be even, and

q

must be odd

Both

p

and

q

must be even

Show answer & explanation

Answer: $p$ must be odd, but $q$ can be any integer

Answer

p

must be odd, but

q

can be any integer

Multiplying any integer

q

by 2 guarantees that

2q

is even. For the sum

3p + 2q

to be odd,

3p

must be odd because the sum of an odd integer and an even integer is odd. Since 3 is odd,

3p

is odd if and only if

p

is odd. The variable

q

can be any integer because

2q

remains even whether

q

is even or odd.

Step-by-Step Solution

1

Analyze the parity of the term

2q

For any integer

q

,

2q

is a multiple of 2, so

2q

is always an even integer.

The product of an even integer and any integer is always even.

2

Determine the required parity of the term

3p

Since

3p + 2q

is odd and

2q

is even,

3p

must be odd.

An odd integer plus an even integer yields an odd sum (

\text{odd} + \text{even} = \text{odd}

).

3

Determine the parity of

p

Since

3p

is odd and 3 is odd,

p

must be an odd integer.

The product of two integers is odd if and only if both integers are odd (

\text{odd} \times \text{odd} = \text{odd}

).

4

Evaluate the restrictions on

q

q

can be any integer (even or odd).

Multiplying

q

by 2 forces

2q

to be even regardless of whether

q

is odd or even.

Key Concept

Parity rules under addition and multiplication of integers

Question 152Question →

If $x$ , $y$ , and $z$ are integers, is the expression $x(y + z)$ an odd integer?

(1) $x^2 + y^2 + z^2$ is an odd integer.
(2) $xy + yz + zx$ is an even integer.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Show answer & explanation

Answer: EACH statement ALONE is sufficient.

Answer

EACH statement ALONE is sufficient.

Each statement independently guarantees that the expression

x(y + z)

must be an even integer. Because a definitive 'NO' answer is obtained from each statement independently, each statement alone is sufficient.

Step-by-Step Solution

1

Analyze the target expression

x(y+z)

.

The expression

x(y+z)

is odd if and only if both

x

is odd AND

(y+z)

is odd. For

(y+z)

to be odd, exactly one of

y

or

z

must be odd and the other even.

Establishing the precise condition for the expression to be odd determines what parity combinations are needed.

2

Evaluate Statement (1):

x^2 + y^2 + z^2

is an odd integer.

Since

k^2

has the same parity as

k

,

x+y+z

must be odd. This occurs in two parity distributions:
- Case 1: All three of

x, y, z

are odd. Here,

y+z = \text{odd} + \text{odd} = \text{even}

, so

x(y+z) = \text{odd} \times \text{even} = \text{even}

.
- Case 2: One variable is odd and two are even.
- Subcase 2a:

x

is odd, while

y

and

z

are even. Then

y+z = \text{even} + \text{even} = \text{even}

, so

x(y+z) = \text{odd} \times \text{even} = \text{even}

.
- Subcase 2b:

x

is even, while one of

y,z

is odd and the other is even. Then

x(y+z) = \text{even} \times \text{odd} = \text{even}

.
In all possible cases,

x(y+z)

is even. Thus, the answer to 'Is

x(y+z)

odd?' is a definitive NO. Statement (1) is sufficient.

Testing all valid parity distributions under Statement (1) shows

x(y+z)

can never be odd.

3

Evaluate Statement (2):

xy + yz + zx

is an even integer.

Note that

x(y+z) = xy + zx

. Therefore,

xy + yz + zx = x(y+z) + yz = \text{even}

.
Suppose for contradiction that

x(y+z)

were odd. Then

yz

would also have to be odd (since

\text{odd} + \text{odd} = \text{even}

). For

yz

to be odd, both

y

and

z

must be odd. But if

y

and

z

are both odd, then

y+z

must be even, which forces

x(y+z) = x \times \text{even} = \text{even}

, contradicting our assumption that

x(y+z)

is odd.
Thus,

x(y+z)

cannot be odd under Statement (2); it must be even. The answer is a definitive NO. Statement (2) is sufficient.

Proof by contradiction demonstrates that

x(y+z)

cannot be odd under Statement (2).

Key Concept

Parity rules of sums and products, including proof by case analysis and contradiction.

Question 153Question →

What is the total number of distinct positive prime factors of the integer $60$ ?

2

3

4

6

12

Show answer & explanation

Answer: $3$

Answer

The total number of distinct positive prime factors of

60

is

3

.

To find the number of distinct positive prime factors of

60

, perform prime factorization:

60 = 2^2 \times 3^1 \times 5^1

. The distinct prime numbers in this expression are

2

,

3

, and

5

. Therefore, there are

3

distinct positive prime factors.

Step-by-Step Solution

1

Find the prime factorization of

60

.

60 = 2^2 \times 3^1 \times 5^1

Expressing the integer as a product of prime numbers reveals all of its prime factors.

2

Identify the distinct prime bases in the factorization.

The distinct prime factors are

2

,

3

, and

5

.

Prime factors are the prime numbers that divide the given integer evenly.

3

Count the number of distinct prime factors.

There are

3

distinct prime factors.

Counting the elements in the set

\{2, 3, 5\}

gives

3

.

Key Concept

Prime Factorization and Prime Factors

Estimated Time:45s

Question 154Question →

For how many integers $n$ satisfying $0 \le n \le 100$ is the expression $n^4 + 2^n + n$ an odd integer?

Show answer & explanation

Answer: 1

Answer

There is exactly 1 value of

n

(specifically

n = 0

) in the range

0 \le n \le 100

for which the expression

n^4 + 2^n + n

is an odd integer.

For any integer

n \ge 1

,

n^4

and

n

share the exact same parity (both even or both odd), making

n^4 + n

always even. Additionally, for

n \ge 1

,

2^n

is an even integer. The sum of two even integers is always even, so

n^4 + 2^n + n

is even for all

1 \le n \le 100

. When

n = 0

, however,

2^0 = 1

is odd and

0^4 + 0 = 0

is even. The sum

0 + 1 = 1

is odd. Thus,

n = 0

is the single value that satisfies the condition.

Step-by-Step Solution

1

Analyze parity of

n^4 + n

n^4 + n

is even for all integers

n

An integer

n

and its power

n^4

always have the same parity. The sum of two integers of the same parity is always even.

2

Evaluate

2^n

parity for

n \ge 1

2^n

is even for all

n \ge 1

Any positive integer power of 2 contains 2 as a prime factor and is therefore even.

3

Evaluate the expression for

n = 0

0^4 + 2^0 + 0 = 1

, which is odd

Zero is a non-negative integer within

0 \le n \le 100

. By definition of exponents,

2^0 = 1

, which is odd.

4

Combine results and count valid integers

Total count = 1

For

n \ge 1

, even + even = even. For

n = 0

, even + odd = odd. Hence, only

n = 0

produces an odd integer.

Key Concept

Parity Rules for Exponent Base 2 Edge Cases and Algebraic Expressions

Question 155Question →

If $m$ and $n$ are integers such that $5m + 3n$ is an even integer and $m - 2n$ is an odd integer, which of the following expressions must be an even integer?

mn + 2

m + n + 1

m^2 + n^2

mn + n + 1

2m + n

Show answer & explanation

Answer: $m^2 + n^2$

Answer

The expression

m^2 + n^2

must be an even integer.

Deduce the parities of

m

and

n

:

5m + 3n = 2(2m + n) + (m + n)

, so

m + n

must be even, implying

m

and

n

have the same parity. Next,

m - 2n

is odd, and since

2n

is even,

m

must be odd. Therefore,

n

is also odd. Evaluating

m^2 + n^2

with

m

and

n

both odd yields

\text{odd}^2 + \text{odd}^2 = \text{odd} + \text{odd} = \text{even}

.

Step-by-Step Solution

1

Analyze the parity of the expression

5m + 3n

.

m + n

is an even integer.

Rewrite

5m + 3n

as

2(2m + n) + (m + n)

. Since

2(2m + n)

is always even,

5m + 3n

has the same parity as

m + n

. Because

5m + 3n

is even,

m + n

must be even, meaning

m

and

n

share the same parity (both even or both odd).

2

Analyze the parity of the expression

m - 2n

.

m

is an odd integer.

Since

2n

is always even,

m - 2n

has the same parity as

m

. Given that

m - 2n

is odd,

m

must be odd.

3

Determine the parity of

n

and evaluate the options.

Both

m

and

n

are odd integers, so

m^2 + n^2 = \text{odd} + \text{odd} = \text{even}

.

Since

m

is odd and

m + n

is even,

n

must also be odd. The square of an odd integer is odd, so

m^2

and

n^2

are both odd, making their sum

m^2 + n^2

an even integer.

Key Concept

Parity rules for integer addition, subtraction, and multiplication

Estimated Time:1m 30s

Question 156Question →

For any integer $k$ , the expression $k(k + 1)(k + 5)$ must be an even integer.

True

False

Show answer & explanation

Answer: True

Answer

True. The expression

k(k + 1)(k + 5)

is guaranteed to be an even integer for all integer values of

k

.

Because

k

and

k + 1

are consecutive integers, one of them must be even. The product of an even integer and any other integer is always even, making

k(k + 1)(k + 5)

even for every integer

k

.

Step-by-Step Solution

1

Analyze the parity of the consecutive terms

k

and

k + 1

.

One of the two integers

k

or

k + 1

is always even regardless of whether

k

is even or odd.

Consecutive integers always alternate between even and odd.

2

Determine the parity of the product of an even integer and any other integer.

The product

k(k + 1)

is always an even integer.

The product of an even integer and any integer is always even.

3

Evaluate the full expression

k(k + 1)(k + 5)

.

Since

k(k + 1)

is even, multiplying by

(k + 5)

produces an even integer.

An even integer multiplied by any integer results in an even integer.

Key Concept

Parity of products of consecutive integers

Estimated Time:1m 0s

Question 157Question →

If $x$ , $y$ , and $z$ are integers such that $(x + 1)(y + 2)$ is an odd integer and $y(z + 3)$ is an even integer, which of the following expressions MUST be an even integer?

x + y + z

xy + z

(y + z)^2 + 1

xz + y

(x + 1)yz

Show answer & explanation

Answer: $x + y + z$

Answer

The expression

x + y + z

MUST be an even integer.

The condition

(x + 1)(y + 2) = \text{odd}

implies both

(x + 1)

and

(y + 2)

are odd, so

x

is even and

y

is odd. Then

y(z + 3) = \text{even}

with

y

being odd requires

(z + 3)

to be even, so

z

is odd. Summing an even integer

x

and two odd integers

y

and

z

yields

\text{even} + \text{odd} + \text{odd} = \text{even}

. Thus, the expression representing the sum of

x

,

y

, and

z

must always be even.

Step-by-Step Solution

1

Determine the parity of

x

and

y

from the first given condition.

x

is even and

y

is odd.

The product

(x + 1)(y + 2)

is odd, which means both factors

(x + 1)

and

(y + 2)

must be odd. If

x + 1

is odd, then

x

is even. If

y + 2

is odd, then

y

is odd.

2

Determine the parity of

z

using the parity of

y

and the second given condition.

z

is odd.

The product

y(z + 3)

is even. Since

y

is odd,

(z + 3)

must be even for the product to be even. If

z + 3

is even, then

z

must be odd.

3

Evaluate the parity of

x + y + z

.

x + y + z

is even.

Substituting the parities gives

\text{even} + \text{odd} + \text{odd} = \text{even} + \text{even} = \text{even}

.

Key Concept

Parity rules for addition and multiplication of integers

Estimated Time:2m 0s

Question 158Question →

Let $f(n) = n^5 + 4n^3 + 3n + 1$ for any positive integer $n$ . For how many integer values of $n$ in the range $1 \le n \le 150$ is the expression $E(n) = (f(n))^n + n^{f(n)}$ an even integer?

Show answer & explanation

Answer: 75

Answer

The total number of integer values of

n

in the given range for which the expression is even is 75.

Analyzing

f(n) \pmod 2

reveals that

n^5 \equiv n \pmod 2

and

3n \equiv n \pmod 2

, making

f(n) \equiv n + 0 + n + 1 \equiv 2n + 1 \equiv 1 \pmod 2

. Since

f(n)

is unconditionally odd,

(f(n))^n

is an odd number raised to a positive integer power, which remains odd. The second term

n^{f(n)}

has positive exponent

f(n) \ge 1

, so its parity matches the base

n

. Therefore,

E(n) = \text{Odd} + n

is even precisely when

n

is odd. In the range

1 \le n \le 150

, exactly half of the 150 integers are odd, yielding 75.

Step-by-Step Solution

1

Evaluate the parity of

f(n)

f(n) = n^5 + 4n^3 + 3n + 1 \equiv n + 0 + n + 1 \equiv 2n + 1 \equiv 1 \pmod 2

, so

f(n)

is always odd.

Powers of an integer retain the base parity (

n^5 \equiv n \pmod 2

), and coefficients modulo 2 simplify terms.

2

Evaluate the parity of

(f(n))^n

and

n^{f(n)}

(f(n))^n

is always odd because an odd base raised to any positive integer exponent is odd.

n^{f(n)}

has the same parity as

n

because

f(n) \ge 1

.

Exponent parity rules state that

\text{odd}^k = \text{odd}

and

\text{even}^k = \text{even}

for positive integer exponents

k

.

3

Formulate the condition for

E(n)

to be even

E(n) = \text{Odd} + n

. Thus,

E(n)

is even if and only if

n

is odd.

The sum of two integers is even if and only if both integers have the same parity (Odd + Odd = Even).

4

Count odd integers in the specified interval

There are 75 odd integers in

\{1, 2, 3, \dots, 150\}

.

Half of the integers in any consecutive sequence of even length starting at 1 are odd.

Key Concept

Odd and Even Integers (Parity)

Estimated Time:2m 0s

Question 159Question →

If $m$ , $n$ , and $p$ are integers such that $m^3 n - n^2 p$ is an odd integer and $m(n + p)$ is an even integer, which of the following expressions MUST be an even integer?

m^2 + n + p

m + np

mn + p

mp + n

n + p + 1

Show answer & explanation

Answer: $m^2 + n + p$

Answer

m^2 + n + p

must be an even integer.

Factoring

m^3 n - n^2 p

as

n(m^3 - np)

shows that

n

is odd and

m^3 - np

is odd. Because

n

is odd,

m^3 - np

has the same parity as

m - p

, meaning

m

and

p

have opposite parities. Testing the condition that

m(n + p)

is even reveals that

p

must be odd and

m

must be even (if

p

were even,

m

would be odd and

m(n + p)

would be odd, a contradiction). With

m

even,

n

odd, and

p

odd, the expression

m^2 + n + p

calculates as

\text{even} + \text{odd} + \text{odd} = \text{even}

, which must be an even integer.

Step-by-Step Solution

1

Analyze the parity of the expression

m^3 n - n^2 p

.

n

is odd and

m^3 - np

is odd.

The expression can be factored as

n(m^3 - np)

. For a product of two integers to be odd, both factors must be odd. Hence,

n

must be odd, and

m^3 - np

must also be odd.

2

Determine the relative parities of

m

and

p

.

m

and

p

must have opposite parities (one is even, the other is odd).

Since

n

is odd,

np

has the same parity as

p

. The expression

m^3 - np

has the same parity as

m - p

. For

m - p

to be odd,

m

and

p

must have opposite parities.

3

Use the second given condition

m(n + p)

is even to determine the exact parity of

m

and

p

.

m

is even and

p

is odd.

If

p

were even, then

n + p = \text{odd} + \text{even} = \text{odd}

. Since

m

and

p

have opposite parities,

m

would be odd, making

m(n + p) = \text{odd} \times \text{odd} = \text{odd}

, which contradicts the condition that

m(n + p)

is even. Thus,

p

cannot be even;

p

must be odd, which implies

m

is even.

4

Evaluate the parity of the options using

m = \text{even}

,

n = \text{odd}

, and

p = \text{odd}

.

m^2 + n + p = \text{even} + \text{odd} + \text{odd} = \text{even}

.

m^2

is even since

m

is even. Summing an even number (

m^2

) and two odd numbers (

n

and

p

) yields an even integer.

Key Concept

Parity rules for integer addition and multiplication: the product of integers is odd if and only if all factors are odd, and the sum of two odd integers is even.

Estimated Time:2m 0s

Question 160Question →

What is the total number of positive factors of the integer $36$ ?

Show answer & explanation

Answer: 9

Answer

The total number of positive factors of

36

is 9.

The prime factorization of

36

is

2^2 \times 3^2

. By the total factors formula, adding 1 to each exponent and multiplying gives

(2+1)(2+1) = 9

. The factors are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Step-by-Step Solution

1

Find the prime factorization of 36

36 = 4 \times 9 = 2^2 \times 3^2

Decomposing the number into prime factors allows the application of the factor count formula.

2

Apply the exponent formula for counting factors

(2 + 1)(2 + 1) = 3 \times 3 = 9

Each factor of

36

is of the form

2^x \times 3^y

, where

x \in \{0, 1, 2\}

(3 choices) and

y \in \{0, 1, 2\}

(3 choices).

Key Concept

Calculating the total number of positive factors from prime factorization exponents.

Estimated Time:45s

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