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If a and b are positive integers such that the greatest common factor of a2 b2 and ab3 is 45, then which of the following could b equal?
  • a)
    3
  • b)
    5
  • c)
    9
  • d)
    15
  • e)
    45
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
If a and b are positive integers such that the greatest common factor ...
One way to solve this problem is to recognize that ab2 is a common factor of a2b2 and ab3. Therefore, it must also be a factor of 45, which can be factored as 5 x 32. Now you can see that b could be 3, and since there can only be one correct answer, A must be it. If you test a = 5 and b = 3, you see that the greatest common factor of 52 x 32 and 5 x 33 is 5 x 32, which equals 45.
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Community Answer
If a and b are positive integers such that the greatest common factor ...

Explanation:

Given:
- The greatest common factor of \(a^2 b^2\) and \(ab^3\) is 45.

Factorization:
- Factorize \(a^2 b^2\) as \(a^2 b^2 = (ab)^2\).
- Factorize \(ab^3\) as \(ab^3 = ab \cdot b^2\).

Prime Factorization:
- Prime factorize \(a^2 b^2 = (ab)^2\) as \(p_1^{2} \cdot p_2^{2}\).
- Prime factorize \(ab^3 = ab \cdot b^2\) as \(p_1 \cdot p_2^{3}\).

Greatest Common Factor (GCF):
- The GCF of \(a^2 b^2\) and \(ab^3\) is the product of the minimum power of each prime factor that appears in both factorizations.
- Since the GCF is 45, it implies that the minimum power of each prime factor is 45.

Analysis:
- For \(a^2 b^2\), the prime factors are \(p_1\) and \(p_2\) raised to the power of 2.
- For \(ab^3\), the prime factors are \(p_1\) and \(p_2\) raised to the power of 1 and 3, respectively.

Conclusion:
- Since the minimum power of each prime factor in the GCF is 45, the prime factors \(p_1\) and \(p_2\) must be raised to the power of 3 and 5, respectively.
- Therefore, \(b\) must equal 3 for the GCF to be 45.
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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 a and b are positive integers such that the greatest common factor of a2 b2 and ab3 is 45, then which of the following could b equal?a)3b)5c)9d)15e)45Correct answer is option 'A'. Can you explain this answer?
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