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Problem Solving Butler: January 2025
January 6PS 1PS 2

________________________________
PS 1

For which of the following numbers is the sum of unique prime factors the maximum?

A. 120
B. 154
C. 170
D. 210
E. 286
________________________________
PS 2

The positive integer P is divisible by 42. The product P x Q is divisible by 180. Which of the following may not be a factor of P x Q ?

A. 630
B. 420
C. 126
D. 108
E. 84
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Data Sufficiency Butler: January 2025
January 7DS 1DS 2

______________________
DS 1

The ratio of the number of women to men to children in a group is 3 : 5 : 9. What is the number of children in the group?

(1) The total number of people in the group is between 60 and 70.
(2) There are more than 15 but less than 25 men in the group.
______________________
DS 2

In the Fortune Frenzy national lottery, a fixed annual cash prize is set initially. For each year that the prize goes unclaimed, an additional $50 million is added to the prize pool for the following year. How much is the fixed annual cash prize set initially?

(1) If the prize goes unclaimed for the first 9 years, the cash prize available for the 10th year will be thrice as large as the prize available for the first year.
(2) If the prize goes unclaimed for the first 5 years, the cash prize available for the 6th year will be $475 million.
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Critical Reasoning Butler: January 2025
January 7CR 1CR 2

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CR 1

In the early 2000s, the incident rate of banking scams in European nations that mandated quarterly bank audits was forty percent higher than that in European nations that have never mandated quarterly bank audits. Thus, one way to bring down the rate of banking scams is to repeal the mandate on quarterly bank audits.

Which of the following, if true, would most weaken the argument above?

A. The annual incident rate of political scams is lower in European nations mandating quarterly bank audits than in European nations without such a mandate.
B. In European nations mandating quarterly bank audits, few banks are penalized for not getting audited every quarter.
C. In European nations without quarterly bank audits, many banks get themselves audited every quarter.
D. The annual incident rate of banking scams in European nations mandating quarterly bank audits has decreased since quarterly bank audits were mandated.
E. Less than half the clients of banks who operate in European nations with quarterly bank audits have their books audited every quarter.
____________________________
CR 2

In 2016, Bob Dylan was awarded the Nobel Prize for Literature for his collective body of songwriting work. However, since songs must be performed in a musical meter, songwriters must ply their craft with an eye to the musicality of their work. For this reason, songwriting is not literature.

Which of the following is an assumption that supports drawing the conclusion above from the reason given for that conclusion?

A. Writing is not literature if its writer pays attention to the musicality of one’s work.
B. Some songwriters are more concerned than others with the musicality of the work they produce.
C. Songwriters should be more concerned with the musicality of their work than they currently are.
D. Some songs are written as an artistic experiment and are not meant to be sung.
E. Writers who produce literature are not concerned with their work’s monetary value.
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Problem Solving Butler: January 2025
January 7PS 1PS 2

__________________________
PS 1

What is the remainder when 131^51 is divided by 11?

A. 1
B. 3
C. 5
D. 7
E. 10
__________________________
PS 2

What is the number of zeroes at the right end in the integer value of 200!/100! ?

A. 0
B. 20
C. 24
D. 25
E. Greater than 25
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Data Sufficiency Butler: January 2025
January 8DS 1DS 2

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DS 1

What was the percentage increase in revenue for company X between 2004 and 2006?

(1) Between 2004 and 2005, the increase was 60% and between 2005 and 2006, the increase was 25%.
(2) Revenue was $100 million in 2003 and $400 million in 2006. Revenue grew by the same amount every year during this period.

_________________________
DS 1

If it rained for twice as long on Monday as on Tuesday, and if it rained for 4 times as long on Wednesday as it did on Monday, how long did it rain over these three days- Monday through Wednesday?

(1) It rained for 30 minutes more on Monday than on Tuesday.
(2) It rained for 3.5 hours longer on Wednesday than on Tuesday.
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Critical Reasoning Butler: January 2025
January 8CR 1CR 2

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CR 1

Last month, four industrial printing presses manufactured by Hawthorne Heavy Machinery and owned by Scarlet Ink Publications broke down in an industrial accident. In response, Scarlet Ink Publications ordered four new printing presses as replacements, from Hawthorne. This decision surprised many in the publishing industry because, ordinarily when a product fails repeatedly, users do not buy that product again.

Which of the following, if true, provides the best indication that Scarlet Ink Publishing’s decision was logically well supported?

A. Although only one of Hawthorne Heavy Machinery’s printing presses broke down during the previous year, the firm’s competitors had a perfect safety record in the same period.
B. Hawthorne Heavy Machinery had to replace many of its employees because they went to work for a competing firm.
C. The Occupational Safety and Health Administration issued new guidelines for printers in order to standardize the safety requirements that govern industrial maintenance.
D. The trade union pressured two major publishers into purchasing safer printing presses so that workers would be safer while operating the machines.
E. The Hawthorne Heavy Machinery printing presses broke down due to operator error, but because of the excellent quality of the printing presses, the damage to the facilities was minimal.
_______________________________
CR 2

Some hotels in California cater almost exclusively to visiting business executives and government officials and rarely have bookings by families with children. Yet, these hotels also have strong ties with companies specializing in entertaining young children.

Which of the following, if true, best reconciles the seeming discrepancy described above?

A. The companies specializing in entertaining young children buy their materials from distributers outside of California.
B. For several weeks during the summer, these hotels become popular destinations for daylong parties hosted by large organizations for employees and their families.
C. Most guests who stay at these hotels prefer to seek entertainment elsewhere rather than at the hotel itself.
D. The few families that stay at these hotels come in groups of four to five, are friendly with one another, and share the cost of the stay.
E. Entertainment provided for young children by outside companies is as enjoyable as that provided by the hotel.
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Problem Solving Butler: January 2025
January 8PS 1PS 2

_____________________________
PS 1

What are the last two digits (at tens place and units place) in the numerical value of 49^852 + 11^77 ?

A. 52
B. 60
C. 72
D. 82
E. 92

_____________________________
PS 2

Suppose that x = 1/2 is the only real number that satisfies \(3^{(-2x^2 + 4Ax - \frac{9}{5})}=√27\). What is the value of A?

A. 19/5
B. 19/10
C. 19/20
D. 53/10
E. 53/20
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Data Sufficiency Butler: January 2025
January 9DS 1DS 2

___________________________
DS 1

Last year, in Crossia, 25% of all babies were delivered in a private hospital, while 40% of all cesarean births occurred in a private hospital. Last year, in Crossia, what percentage of all births that took place in a private hospital were cesarean births?

(1) Last year, in Crossia, a total of 72,000 babies were delivered in a private hospital.
(2) Last year, in Crossia, 20% of all births were caesareans.
___________________________
DS 2

What is the average (arithmetic mean) mileage of 7 cars in a parking lot?

(1) If the 7 mileages were ordered from least to most, each would be 2.5 miles per gallon higher than the preceding car’s mileage.
(2) The range of the 7 mileages is 15 miles per gallon.
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Critical Reasoning Butler: January 2025
January 9CR 1CR 2

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CR 1

Due to certain persistent limitations in the technology, many early buyers of virtual reality systems apparently lost interest in buying new virtual reality games to run on their systems. The sale of virtual reality games is still going strong because the number of homes with virtual reality systems is still growing. However, businesses distributing virtual reality games will face hard times because the market for virtual reality systems is likely to saturate soon.

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

A. The market for virtual reality systems is considered saturated only after seventy percent of all homes have at least one such system.
B. Early buyers of novel products are always quicker to acquire novelties, but also quicker to tire of them than other buyers.
C. Customer satisfaction surveys reveal that few of the early buyers of virtual reality systems had any complaints regarding this new product’s performance aspects.
D. Several games specifically designed to be played in virtual reality form a sizeable proportion of all the products handled by retailers of virtual reality games.
E. Shrinking markets generally intensify competition among businesses, leading to the failure of several marginal businesses.
___________________________
CR 2

At the time of the last census, the number of military veterans was slightly greater than the number of conservatives. Military veterans have tripled in number since then, while conservatives have increased in number, but at a lower rate than have military veterans.

Which of the following must be true if the statements above are true?

A. There are now fewer military veterans than there are conservatives.
B. The ratio between military veterans and conservatives has increased since the last census.
C. The ratio between military veterans and conservatives has decreased since the last census.
D. Every conservative is also a military veteran.
E. The ratio between military veterans and conservatives has not changed since the last census.
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Problem Solving Butler: January 2025
January 9PS 1PS 2

________________________
PS 1

If N is a positive integer and N! is divisible by 143, what is the lowest possible value of N ?

A. 10
B. 11
C. 13
D. 142
E. 143
________________________
PS 2

If n is a positive integer and n^2 is divisible by 120, which is the highest positive integer that must divide n ?

A. 24
B. 30
C. 60
D. 90
E. 120
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Problem Solving Butler: January 2025
January 12PS 1PS 2

_________________________
PS 1

A and B are integers greater than 1 and A^3 x B is a multiple of 10. Which of the following must be a multiple of 8,000 ?

A. A*B^3
B. A^3 * B^3
C. A^3 * B^6
D. A^6 * B^3
E. A^6 * B^6

_________________________
PS 2

What are the last two digits (at tens place and units place) in the numerical value of the following product?

7249 x 5772 x 5313 x 7775 x 8263 x 317 x 9937

A. 00
B. 10
C. 30
D. 50
E. 90
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