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Study Plan Q&A Chat 22 Jan 2026, 01:52
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Problem Solving Butler: January 2025
January 22PS 1PS 2

_________________________
PS 1

What is the remainder when 21^32 is divided by 50?

A. 1
B. 11
C. 21
D. 31
E. 41

_________________________
PS 2

If P is a positive integer, which of the following is not possible for any value of x?

A. \(-(P^x) = (-P)^{(-x)}\)

B. \(-(P^x) = (-P)^x\)

C. \((P)^{(-x)} = -(P)^x\)

D. \((-P)^x = (-P)^{(-x)}\)

E. \((-P)^x = -(P^x)\)
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Study Plan Q&A Chat 22 Jan 2026, 01:54
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Critical Reasoning Butler: January 2025
January 22CR 1CR 2

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CR 1

Salmon meat is frequently used in gourmet cooking and retails at high prices. Meat from less valuable white fish, however, can be dyed to give it salmon’s distinct red/pink hue. Only experienced chefs are able to distinguish between genuine salmon and counterfeit salmon prepared in this manner. However, only ten percent of white fish caught can be dyed and much of the meat from these fish is not of a high enough quality for making the treatment worthwhile. Thus, surely the vast majority of salmon sold is genuine.

A serious flaw in the reasoning of the argument is that

A. comparisons between the price of salmon sold for gourmet cooking and its value for personal consumption are not mentioned
B. it does not consider the difficulty that a chef would have in distinguishing between genuine salmon and dyed white fish
C. the possibility that white fish may be used in gourmet cooking, even if it has not been dyed is overlooked
D. it treats the currently available method of dying white fish to resemble salmon as if it were the only method that can be used to do so
E. it does not consider how rare white fish dyed red/pink is in comparison with how rare genuine salmon is

____________________________
CR 2

The participants sorted into the experimental group of a consumer focus test were instructed to use a new brand of degreasing agent to clean their car engines for a month. Afterward, the members of this group reported a drop in their cars’ acceleration speed, while the control group, those who did not use the degreasing agent, did not report any drop. The detrimental effects on the cars were attributed to a petroleum solvent that is the principal active ingredient in the degreasing agent.

Which of the following, if true, would best help explain how the degreasing agent might produce the observed effect?

A. Chemical analysis of the degreasing agent determined that a concentrated form of the agent was used.
B. Subjects in the focus group did not know whether they were cleaning their car engines with the new degreasing agent or some other substance that is harmless.
C. As the primary use of the degreasing agent is as a cleaning agent, its negative effects are rarely noticed by car owners.
D. The petroleum solvent that is a constituent of the degreasing agent is also sold separately as a heavy-duty industrial cleaner.
E. A high level of the petroleum solvent degrades engine components responsible for the acceleration process.
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Study Plan Q&A Chat 22 Jan 2026, 01:57
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Data Sufficiency Butler: January 2025
January 22DS 1DS 2

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DS 1

Anne is participating in a shooting event. If she shoots five times, what is the probability that she hits the bull’s eye at least once?

(1) The probability of her hitting the bull’s eye is 20% in each shoot.
(2) The probability of her not hitting the bull’s eye is 80% in each shoot.

______________________
DS 2

Company M has two divisions – X and Y. Both the divisions have exactly two types of employees – operators and executives. Is the total number of employees in Division X greater than that in division Y?

(1) The division X has 10% executives and division Y has 25% executives.
(2) The company M has 15% executives.
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Study Plan Q&A Chat 23 Jan 2026, 00:41
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Problem Solving Butler: January 2025
January 23PS 1PS 2

______________________
PS 1

Which of the following is not an integer ?

A.\(\frac{25! }{ (5)^6}\)

B. \(\frac{10! }{ (3)^4}\)

C. \(\frac{9! }{ (2)^7}\)

D. \(\frac{12! }{ (3!)^5}\)

E. \(\frac{20! }{ (11)^2}\)
______________________
PS 2

How many factors of 72^2 are multiples of 6 ?

A. 30
B. 24
C. 20
D. 15
E. 5
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Study Plan Q&A Chat 23 Jan 2026, 01:09
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Critical Reasoning Butler: January 2025
January 23CR 1CR 2

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CR 1

In an interview to a business daily, a senior manager at Windsor Hotels recently stated that the hotel is poorly managed, mentioning the hotel’s inability to modify its marketing strategy in response to a recent surge in its number of unoccupied rooms. A senior manager’s negative comments are generally detrimental to the image of the company, but in this case, the negative comments were also unjustified. Although an increased number of unoccupied rooms at a hotel often indicates that the hotel is losing clients to competitors, in Windsor Hotels’ situation, it does not. Rather, the increase in the number of unoccupied rooms can be attributed entirely to the cancellation, earlier this year, of two of the city’s biggest events.

In the argument given, the two boldfaced portions play which of the following roles?

(A) The first states the position opposed by the argument as a whole; the second states the conclusion of the argument as a whole.
(B) The first states the position opposed by the argument as a whole; the second provides evidence to undercut the support for the position being opposed.
(C) The first states the position opposed by the argument as a whole; the second is evidence that has been used to support the position being opposed.
(D) The first is evidence that has been used to support a position opposed by the argument as a whole; the second provides information to undercut the force of that evidence.
(E) The first is evidence that has been used to support a position opposed by the argument as a whole; the second states the conclusion of the argument as a whole.
_____________________
CR 2

To compete with a number of newer restaurants, Rick’s Café has begun offering a free appetizer to every customer who orders more than $40 worth of food. The manager of Rick’s Café claims that the offer has been a huge success in increasing sales, since the number of customers whose orders exceed $40 has increased by twenty-seven percent since the offer began.

Which of the following, if true, most seriously weakens the claim of the manager?

A. Most people who began eating at Rick’s Café after the offer began spending more than $40 each time they eat there.
B. Most people who typically spend less than $25 on a restaurant meal would not be persuaded to spend more by any offer.
C. Nearly all who have begun spending $40 or more at Rick’s Café since the offer began are regular patrons who have increased the average amount spent during one meal at Rick’s Café, but have begun visiting less frequently.
D. Since the offer began, most of the people who spend $40 or more at Rick’s Café are people who have never eaten there before and who always spend more than $40 when eating at restaurants.
E. Most people who received a free appetizer through the offer will continue to eat at Rick’s Café after the offer ends.
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Study Plan Q&A Chat 23 Jan 2026, 01:11
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Data Sufficiency Butler: January 2025
January 23DS 1DS 2

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DS 1

If 40 percent of the people who bought a particular smartphone bought a smartwatch, what percent of the smartphone buyers bought headphones?

(1) 50 percent of people who bought the smartphone and the smartwatch also bought headphones.
(2) 80 percent of people who bought the smartphone and the headphones also bought the smartwatch.
__________________________
DS 2

Over the course of a certain month, Olga, a saleswoman, sold 20 cars, and each of her 7 colleagues sold at least 1 car. Did Olga sell as many or more cars than at least 4 of her colleagues?

(1) The average (arithmetic mean) of the number of cars sold by Olga’s 7 colleagues is 12.
(2) The median of the number of cars sold by Olga’s 7 colleagues is 18.
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Study Plan Q&A Chat 25 Jan 2026, 22:50
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Problem Solving Butler: January 2025
January 26PS 1PS 2

__________________
PS 1

If \((\frac{1}{50})^a(\frac{1}{20})^4 = \frac{1}{50} (100)^{-6}\), a = ?

A. 3
B. 4
C. 5
D. 6
E. 7

__________________
PS 2

What is the highest integer k for which 125^k perfectly divides 200! ?

A. 49
B. 48
C. 16
D. 13
E. 1
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Study Plan Q&A Chat 25 Jan 2026, 22:53
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Critical Reasoning Butler: January 2025
January 26CR 1CR 2

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CR 1

A new town ordinance that requires all home owners to install burglar alarms and other anti-theft measures on their property has received stiff opposition from those who believe that citizens of a free society must be allowed to freely take risks, as long as those risks do not harm others. Resultantly, they conclude that it should be each home owner’s decision to install the anti-theft measures.

Which of the following, if true, most seriously weakens the conclusion drawn above?

A. Many new homes come with pre-installed anti-theft measures.
B. Property owners who do not install anti-theft measures, on average, experience greater monetary loss due to burglary than those who do install such measures.
C. Banks are required to install extremely stringent anti-theft measures.
D. The rate of burglary in states that do not have laws mandating the installation of anti-theft measures is greater than that in states that do have such laws.
E. Theft insurance rates are higher for everyone because of the need to pay for the increased losses sustained by those without anti-theft measures.
_________________________
CR 2

Lunacorp and Salem Industries are two sanitation firms that employ an equal number of workers who work the same number of hours per week. According to each company’s end of year report, the employees of Lunacorp took fewer sick leaves last year than did the employees of Salem Industries. Therefore, employees of Lunacorp are less likely to fall ill than are employees of Salem Industries.

Which of the following, if true, would most strengthen the conclusion above?

A. Lunacorp’s expenditure on new job-related medical claims is greater than Salem Industries’.
B. Salem Industries holds more health inspections than does Lunacorp.
C. Salem Industries’ infirmary is more modern than Lunacorp’s.
D. Salem Industries provides sanitation services for environments that are more hazardous for its employees to work in than does Lunacorp.
E. Salem Industries provides more types of health-care benefits than does Lunacorp.
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Study Plan Q&A Chat 25 Jan 2026, 22:56
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Data Sufficiency Butler: January 2025
January 26DS 1DS 2

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DS 1

Lily has only red and blue balls in a jar. If she removes three balls, is the probability of getting all red balls greater than the probability of getting at least one blue ball?

(1) The number of red balls is more than three times the number of blue balls.
(2) Less than 1 / 4 th of the balls in the jar are blue.

___________________________
DS 2

How many people did Casax Inc. employ in 2008?

(1) In 2007, Casax Inc.’s workforce numbered 100,000 people.
(2) In 2008, Casax Inc.’s workforce increased by 2 percent, or 2,000 people, over 2007.
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Study Plan Q&A Chat 26 Jan 2026, 22:51
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Problem Solving Butler: January 2025
January 27PS 1PS 2

____________________
PS 1

K is a factor of 30! + 40!. If K is a prime number less than 40, the highest possible value of K is?

A. 19
B. 23
C. 29
D. 31
E. 37
____________________
PS 2

If y = 80x + 36, where x is a positive integer, which of the following cannot be factor of y ?

A. 3
B. 7
C. 12
D. 14
E. 24
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Study Plan Q&A Chat 26 Jan 2026, 22:54
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Critical Reasoning Butler: January 2025
January 27CR 1CR 2

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CR 1

Technician: It is true that the Nebula G10 smartphone is not as valuable as the Outlook M1 smartphone because, on average, insurance companies do not pay as much to replace a lost Nebula G10 as to replace a lost Outlook M1. Repair costs show a different picture though. It can’t be denied that the repairs to Nebula G10 phones are especially labor-intensive and labor is a significant factor in accident repair costs. As a result, the average amount that insurance companies pay to repair a phone under warranty that is damaged by an accident is typically higher for the Nebula G10 than for the Outlook M1.

Which of the following, if true, most strongly supports the technician’s explanation?

A. Reliable statistics show that Outlook M1 phones are more frequently damaged in accidents than Nebula G10 phones are.
B. Outlook M1 phones are considered more economically valuable largely for the ease of repair of the phones.
C. Availability of technicians specializing in the repairs of Nebula G10 is higher than that of technicians specializing in the repairs of Outlook M1.
D. No significant difference exists between the cost of routine upkeep of Outlook M1 and that of Nebula G10.
E. Consumer data indicates that the Nebula G10 phones are returned for defects at a rate greater than that for Outlook M1 phones.
_________________________
CR 2

Forensic scientists trained to authenticate antiques do so by analyzing not only the physical form of the antique but also such properties as the chemical composition of the materials it is composed of and microscopic signs of damage that indicate its age. Even the most adept forgers cannot duplicate all of the properties the scientists analyze.

Which of the following can be logically concluded from the passage above?

A. The time it takes for the forensic scientists to examine an antique and determine its authenticity makes their services impractical for commercial use.
B. Nobody can secure authentication from a forensic scientist trained to authenticate antiques by relying solely on a skill for forging antiques.
C. Forensic scientists trained to authenticate antiques will soon be hired by most museums.
D. Forensic forgery-detection training has taken many years to formulate and perfect.
E. In many cases even genuine antiques are denied authentication by forensic scientists trained to authenticate antiques.
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Study Plan Q&A Chat 26 Jan 2026, 22:57
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Data Sufficiency Butler: January 2025
January 27DS 1DS 2

_______________________
DS 1

Each of the 600 cars in a certain parking lot is either a hatchback or a sedan. How many of the hatchbacks are electric?

(1) 75% of all cars in the parking lot are electric.
(2) The number of electric sedans is 25% more than the number of electric hatchbacks.
_______________________
DS 2

Last week, Cosmo repaired multiple motorbikes and cars. How many hours did Cosmo spend repairing motorbikes last week?

(1) Last week, Cosmo spent a total of 40 hours repairing both cars and motorbikes.
(2) Last week, the average (arithmetic mean) number of hours that Cosmo spent repairing each motorbike was 20 percent greater than the average number of hours he spent repairing each car.
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Study Plan Q&A Chat 27 Jan 2026, 23:02
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Problem Solving Butler: January 2025
January 28PS 1PS 2

_________________________
PS 1

If \(\frac{9^{x-1}}{27^{3x+2}} = 81^{x+1}\), x = ?

A. 11/12
B. 12/11
C. 4/3
D. -11/12
E. -12/11
_________________________
PS 2

If P is a positive integer and Q = 5P + 7, which of the following CANNOT be a divisor of Q?

A. 2
B. 3
C. 5
D. 7
E. 11
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Study Plan Q&A Chat 27 Jan 2026, 23:05
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Critical Reasoning Butler: January 2025
January 28CR 1CR 2

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CR 1

The introduction of ladybugs into a garden ecosystem is an excellent method of protecting ornamental flowers from aphids. Unfortunately, the ladybugs will only prey on the aphids for a week before flying away, and no aphid infestation severe enough to warrant the deployment of ladybugs can be cleared in a single week. However, although synthetic pesticides are not much longer lasting, they can be easily reapplied whenever the aphids reappear. Clearly, therefore, for protecting ornamental flowers from aphids, introducing ladybugs can play no useful role.

The argument is most vulnerable to the objection that it fails to

A. consider that there might be less severe aphid infestations that can be cleared in a week
B. allow for the possibility of combining the two aphid-control methods it mentions
C. distinguish among the several varieties of synthetic pesticides
D. mention any alternative aphid control method that could be utilized in place of the rejected method
E. allow for the possibility that the introduction of insects other than ladybugs might have the same effect.
__________________________
CR 2

Environmentalist: Cod prices in our area need to be increased significantly. The cod population in the nearby ocean has been falling for years because cod is being fished at unsustainable levels. Due to the low price of cod, few consumers have been affected by the introduction of the hefty environmental tax on the sale of this endangered fish.

The two sections in boldface play which of the following roles in the environmentalist’s argument?

(A) The first is a conclusion that the argument provides support for, and which in turn provides support for the main conclusion; the second is the main conclusion.
(B) The first is the only conclusion; the second provides an explanation for the first.
(C) The first is an observation that the second provides an explanation for; the second is the main conclusion but not the only conclusion.
(D) The first is a premise that supports the main conclusion of the argument; so is the second.
(E) The first is the main conclusion; the second is a conclusion supported by the argument, and which in turn supports the first.
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Study Plan Q&A Chat 27 Jan 2026, 23:09
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Data Sufficiency Butler: January 2025
January 28DS 1DS 2

________________________
DS 1

Is the median monthly salary of 15 randomly selected employees of a particular company greater than the average (arithmetic mean) monthly salary by at least 5 percent?

(1) The median monthly salary of the 15 employees is $250 greater than their average monthly salary.
(2) The sum of the monthly salaries of the 15 employees is less than $70,000.

________________________
DS 2

Jack paid $30 at a salon; this included the salon fee, taxes, and tip. How much tax did Jack pay in the transaction?

(1) Tax and Tip together contributed to 25% of the total expense.
(2) Salon fee and the tip together contributed to 90% of the total expense.
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Study Plan Q&A Chat 28 Jan 2026, 08:38
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If I need to practice ’ValuesOrders/factors’

Is there any other way I can generate insights as to where I was specifically weak in?
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Study Plan Q&A Chat 28 Jan 2026, 23:01
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Data Sufficiency Butler: January 2025
January 29DS 1DS 2

________________________
DS 1

The average selling price per sofa-set for a showroom in 2016 was $40 more than that in 2015. What was the revenue for the showroom, from the sale of sofa-sets, in 2015?

(1) The showroom sold 20% more number of sofa-sets in 2015 than in 2016.
(2) The revenue of the showroom, from the sale of sofa-sets, was $30,000 more in 2015 than in 2016.

________________________
DS 2

A university offers 3 courses in mathematics: a BS program, an MS program, and a dual degree program. The annual tuition fee for the dual degree program is 20% less than the annual tuition fees for the other 2 programs taken together. If 60% of mathematics students in the upcoming class will attend the BS program and the remaining are equally divided between the other 2 programs, which of the 3 programs will generate the most tuition fees for the university in the coming year?

(1) The annual tuition fee for the MS program is 30% higher than that of the BS program.
(2) The annual tuition fee for the BS program is $30,000.
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Study Plan Q&A Chat 28 Jan 2026, 23:04
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Critical Reasoning Butler: January 2025
January 29CR 1CR 2

___________________________
CR 1

Candidate: Today’s voters expect their government to be accountable. Virtually, every improvement in government’s accountability inspires a new round of demands for greater accountability. If a politician is satisfied with the government’s current level of accountability, that politician will soon find that the constituents are not. As your representative in parliament, meeting your demands for accountability will be my goal!

Which of the following must be true on the basis of the statements above?

A. The candidate’s competitors will succeed in attracting voters only if those competitors adopt the candidate’s goals as their own.
B. The candidate’s goal is possible to meet, only if it is possible to continue the current advances in government accountability.
C. A politician who fails to accurately anticipate the demands of the constituents is certain to fail in advancing government accountability.
D. If a politician becomes satisfied with the accountability that the government displays, the government accountability will decline.
E. The candidate’s constituents are currently satisfied with the accountability displayed by the government.
___________________________
CR 2

The World Energy Council maintains ranking of nations, detailing which country’s citizens are the most and least energy conscious by measuring the amount of electricity consumed per every 100 citizens.

Of the following, each statement, if true, would serve as a logical objection to using the council’s ranking as a representation of the energy consciousness of citizens in each nation EXCEPT:

A. The refrigeration of food consumes significantly less electricity in countries with a temperate or arctic climate than it does in countries with a tropical climate.
B. Some countries have hundreds of millions of citizens and some barely have a few millions.
C. The average citizen in developed nations can afford to purchase more energy-efficient technology than can citizens of less developed nations.
D. Due to low electricity production, many developing countries regularly shut down residential electrical supply for as long as a few hours every day for several months in a year.
E. The power grids of some countries are in poor shape and thus waste more electricity than do power grids of other countries.
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Problem Solving Butler: January 2025
January 29PS 1PS 2

_________________________
PS 1

If a, b, c, d, e, and f represent different prime numbers each, which of the following may represent the value of 16! ?

A. a^15 x b^6 x c^3 x d^2 x e^2 x f
B. a^15 x b^5 x c^3 x d^2 x e x f^2
C. a^15 x b^6 x c^3 x d^2 x e x f
D. a^8 x b^5 x c^3 x d^2 x e x f
E. a^8 x b^5 x c^2 x d^2 x e x f
_________________________
PS 2

p, m, and n are positive integers greater than 1 such that p = m x n and m and n do not have any common prime factors. If p has 25 factors, m has r factors, and n has s factors, what is the value of (r + s)?

A. 4
B. 8
C. 10
D. 16
E. 25
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Study Plan Q&A Chat 29 Jan 2026, 22:45
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Problem Solving Butler: January 2025
January 30PS 1PS 2

_______________________________
PS 1

If 3p^5 = 96 and 7q = 32, what is the value of p/q?

A. 16/7
B. 7/16
C. ±16/7
D. ±7/16
E. Cannot be determined from the given information
_______________________________
PS 2

p, q, r, s are positive integers. When p is divided by q, the remainder is 7; when r is divided by s, the remainder is 12. Which of the following may be a value of q + s?

A. 15
B. 17
C. 19
D. 20
E. 23
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