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ACT Exam  >  ACT Questions  >  If x and y are positive integers such that th... Start Learning for Free
If x and y are positive integers such that the greatest common factor of x2y2 and xy3 is 27, then which of the following could y equal?
  • a)
    81
  • b)
    27
  • c)
    18
  • d)
    9
  • e)
    3
Correct answer is option 'E'. Can you explain this answer?
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Verified Answer
If x and y are positive integers such that the greatest common factor ...
The correct answer is E. You could take a “bruteforce” approach and test all the given values of y and see if you could find an x that worked. For aenxdamxple, if y = 9, then the two numbers are x2 × 92 × 93. You can see that 92 is a factor of these 2 numbers, so 27 cannot be the greatest common factor.
It might be more efficient to be more general and avoid testing all 5 values of x. Notice that xy2 is a common factor of both x2y2  and xy3. Because it is a factor, xy2 must also be a factor of 27. Well, 27 factors as 3 × 32, so it seems natural to see = 3 and y = 3 are possible solutions. In this case the two numbers from the problem are 32 × 32 and 33 × 3 and the g
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Most Upvoted Answer
If x and y are positive integers such that the greatest common factor ...
To find the possible values of y, we need to analyze the given information about the greatest common factor (GCF) of x^2y^2 and xy^3.

- Given: GCF(x^2y^2, xy^3) = 27

Let's break down the expressions x^2y^2 and xy^3 and examine their factors.

Expression 1: x^2y^2
- Factors of x^2y^2 include x, x, y, and y.

Expression 2: xy^3
- Factors of xy^3 include x, y, y, and y.

Now, let's consider the GCF of these two expressions.

- GCF(x^2y^2, xy^3) = 27

Since the GCF is 27, all the common factors between the two expressions must be raised to a power of 27. However, we also know that x and y are positive integers.

From the factors mentioned above, we can see that the only common factor of x^2y^2 and xy^3 is y. Thus, y must be raised to the power of 27.

Now, let's consider the given answer choices.

a) 81: This is not a possible value for y because 81 is not a power of 27.
b) 27: This is not a possible value for y because 27 is not a power of 27.
c) 18: This is not a possible value for y because 18 is not a power of 27.
d) 9: This is not a possible value for y because 9 is not a power of 27.
e) 3: This is a possible value for y because 3^3 = 27.

Hence, the correct answer is option E, y = 3.
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If x and y are positive integers such that the greatest common factor ...
The correct answer is E. You could take a “bruteforce” approach and test all the given values of y and see if you could find an x that worked. For aenxdamxple, if y = 9, then the two numbers are x2 × 92 × 93. You can see that 92 is a factor of these 2 numbers, so 27 cannot be the greatest common factor.
It might be more efficient to be more general and avoid testing all 5 values of x. Notice that xy2 is a common factor of both x2y2  and xy3. Because it is a factor, xy2 must also be a factor of 27. Well, 27 factors as 3 × 32, so it seems natural to see = 3 and y = 3 are possible solutions. In this case the two numbers from the problem are 32 × 32 and 33 × 3 and the g
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Directions:Read the passage and choose the best answer to each question.PassageCoronary heart disease affects millions of people worldwide each year. It is the end result of a build up of plaque (cholesterol) on the interior walls of the arteries that supply the muscle of the heart. Most individuals with coronary heart disease show no evidence of disease for decades as the disease progresses before the first onset of symptoms, often a “sudden” heart attack. After decades of building up on the artery walls, the plaque may reduce the blood flow to the heart muscle. There are several hypotheses that have been proposed to explain the causes of plaque build-up leading to coronary heart disease.Behavioral HypothesisThe primary causes of coronary heart disease are behavioral factors such as diet, risky behaviors, and level of physical activity. Coronary heart disease is associated with smoking, obesity, hypertension (chronic high blood pressur e) and a lack of vitamin C. According to one study, individuals who consume large amounts of saturated fats and trans-fats have high levels of cholesterol and are at higher levels of risk for heart disease. Vegetarians have been shown to have a 24% reduced risk of heart disease due to dietary modifications alone. In addition, extra weight is thought to lead to higher total cholesterol levels, high blood pressure, and an increased risk of coronary heart disease. Obesity increases the chances of developing other risk factors for heart disease, especially high blood pressure, high blood cholesterol, and diabetes. Smoking is also a major cause of heart disease as it puts individuals at higher risk of developing a number of chronic disorders. Furthermore, people who are not physically active have a greater risk of heart attack than do people who exercise regularly. Exercise burns calories, helps to control cholesterol levels and diabetes, and may lower blood pressure. Exercise also strengthens the heart muscle and makes the arteries more flexible.Familial HypothesisCoronary heart disease is genetically inherited, meaning it tends to run in families. For example, people whose parents or siblings had a heart or circulatory problem before the age of 55 are at greater risk for heart disease than someone who does not have that family history. Risk factors (including hypertension, diabetes, and obesity) may also be passed from one generation to another. Studies have determined that the single greatest indicator of risk for coronary heart disease is family history. Other studies have shown isolated populations to be significantly more or less susceptible to coronary heart disease than is the general population. Isolated populations share the same gene pool (a set of genetic traits found within a population), which supports the proposition that family history is the primary cause and indicator of the disease.Q.Assume that increased cholesterol levels result in increased risk for coronary heart disease. How would supporters of the Familial Hypothesis explain the study cited in the Behavioral hypothesis?

Directions:Read the passage and choose the best answer to each question.PassageCoronary heart disease affects millions of people worldwide each year. It is the end result of a build up of plaque (cholesterol) on the interior walls of the arteries that supply the muscle of the heart. Most individuals with coronary heart disease show no evidence of disease for decades as the disease progresses before the first onset of symptoms, often a “sudden” heart attack. After decades of building up on the artery walls, the plaque may reduce the blood flow to the heart muscle. There are several hypotheses that have been proposed to explain the causes of plaque build-up leading to coronary heart disease.Behavioral HypothesisThe primary causes of coronary heart disease are behavioral factors such as diet, risky behaviors, and level of physical activity. Coronary heart disease is associated with smoking, obesity, hypertension (chronic high blood pressur e) and a lack of vitamin C. According to one study, individuals who consume large amounts of saturated fats and trans-fats have high levels of cholesterol and are at higher levels of risk for heart disease. Vegetarians have been shown to have a 24% reduced risk of heart disease due to dietary modifications alone. In addition, extra weight is thought to lead to higher total cholesterol levels, high blood pressure, and an increased risk of coronary heart disease. Obesity increases the chances of developing other risk factors for heart disease, especially high blood pressure, high blood cholesterol, and diabetes. Smoking is also a major cause of heart disease as it puts individuals at higher risk of developing a number of chronic disorders. Furthermore, people who are not physically active have a greater risk of heart attack than do people who exercise regularly. Exercise burns calories, helps to control cholesterol levels and diabetes, and may lower blood pressure. Exercise also strengthens the heart muscle and makes the arteries more flexible.Familial HypothesisCoronary heart disease is genetically inherited, meaning it tends to run in families. For example, people whose parents or siblings had a heart or circulatory problem before the age of 55 are at greater risk for heart disease than someone who does not have that family history. Risk factors (including hypertension, diabetes, and obesity) may also be passed from one generation to another. Studies have determined that the single greatest indicator of risk for coronary heart disease is family history. Other studies have shown isolated populations to be significantly more or less susceptible to coronary heart disease than is the general population. Isolated populations share the same gene pool (a set of genetic traits found within a population), which supports the proposition that family history is the primary cause and indicator of the disease.Q.One advantage of the Behavioral Hypothesis is that it best explains why heart disease is more common in which of the following groups?

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If x and y are positive integers such that the greatest common factor of x2y2 and xy3 is 27, then which of the following could y equal?a)81b)27c)18d)9e)3Correct answer is option 'E'. Can you explain this answer?
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