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Hi
tgubbay1,
Why don't you give an attempt on this practice question:
Q: \(x, y\) are positive integers such that \(x\) is the lowest number which has 15 and 20 as its factor. Is \(x\) a factor of \(y\)?
(1) \(\frac{y}{2 *3^2}\) is an integer.
(2) \(\frac{y * z}{5}\) is an integer where \(z\) is a positive even integer whose units digit is not 0.
Let me know your approach and analysis for this practice question.
Also, you may try your hand at this question which tests similar concepts
if-x-is-a-positive-integer-is-x-1-a-factor-of-126421.htmlRegards
Harsh
Detailed Solution:Step-I: Given Info We are given two positive integers \(x\) and \(y\) such that \(x\) is the lowest number that has 15 & 20 as its factors. We are asked to find if \(x\) is a factor of \(y\).
Step-II: Interpreting the Question StatementWe are asked to find if \(x\) is a factor of \(y\). For \(x\) to be a factor of \(y\), \(y\) should have the prime factors of \(x\) in their respective powers as its factors.
We are given that \(x\) is the lowest number that has 15 & 20 as its factors. We know that the lowest number which has two numbers as its factor is the LCM of those numbers. Prime factorizing 15 and 20 would give us
\(15 = 3^1 * 5^1\) and \(20 = 2^2 * 5^1\). We know that for calculating LCM we take the highest powers of prime factors. So, LCM (15, 20) = \(2^2 * 3^1 * 5^1\)
\(x=2^2*3^1 * 5^1\) . For \(x\) to be a factor of \(y\), \(y\) should have \(2^2, 3^1\) and \(5^1\)as its factors. Let’s see if the statements provide us sufficient information about the prime factors of \(y\).
Step-III: Analyze Statement-I independentlyStatement- I tells us that \(\frac{y}{2*3^2}\) is an integer i.e. \(y\) has \(2\), \(3^2\) as its factors. For \(x\) to be a factor of \(y\), \(y\) should have \(2^2, 3^1\) and \(5^1\) as its factors. We don’t know about the other prime factors of \(y\).
So, statement-I is not sufficient to answer the question.
Step-IV: Analyze Statement-II independently Statement-II tells us that \(\frac{y*z}{5}\) is an integer where \(z\) is a positive even integer whose units digit is not 0. Since, \(z\) is a positive even integer whose units digit is not 0, we can say that \(z\) is not a multiple of 5. Hence if \(\frac{y*z}{5}\) is an integer with \(z\) not being a multiple of 5, that would mean that \(y\) is a multiple of 5.
But we don’t know if \(y\) has \(2^2, 3^1\) as its factors.
So, statement-II is not sufficient to answer the question
Step-V: Combining Statements I & IICombining statement- I & II, we can see that \(y\) has \(2,3^2,5\) as its factors. For a moment, combining both the statements seem sufficient to answer the question. But if we look back at step-II, we can see that for \(x\) to be a factor of \(y\), \(y\) should have \(2^2, 3^1\) and \(5^1\) as its factor. The power of the prime factor\(2\) is not sufficient to tell us if \(x\) is a factor of \(y\). Hence, we can’t say if \(x\) is a factor of \(y\).
So, combining both the statements is also not sufficient to answer the question.
Answer: (E) Key takeawaysFor most of the LCM-GCD questions, prime factorization is the key to the solution.To know more in detail about the likely mistakes in LCM-GCD question, read the article
3 mistakes in LCM-GCD questionsRegards
Harsh