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What is the value of number X?

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What is the value of number X?  [#permalink]

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New post Updated on: 13 Aug 2018, 01:59
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e-GMAT Question:



What is the value of number X?

1) The HCF of X and 36 is 4
2) The LCM of X and 36 is 72


    A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
    B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
    C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
    D) EACH statement ALONE is sufficient.
    E) Statements (1) and (2) TOGETHER are NOT sufficient.

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Question 3 of The e-GMAT Number Properties Marathon




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Question 4 of the Marathon


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Originally posted by EgmatQuantExpert on 27 Feb 2018, 10:00.
Last edited by EgmatQuantExpert on 13 Aug 2018, 01:59, edited 3 times in total.
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Re: What is the value of number X?  [#permalink]

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New post 27 Feb 2018, 11:31
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EgmatQuantExpert wrote:

Question:



What is the value of number X?

1) The HCF of X and 36 is 4
2) The LCM of X and 36 is 72


    A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
    B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
    C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
    D) EACH statement ALONE is sufficient.
    E) Statements (1) and (2) TOGETHER are NOT sufficient.


Statement 1: \(36=2^2*3^2\). Hence \(x=4k\), where \(k\) is any integer. Insufficient

Statement 2: \(72=2^3*3^2\). Hence \(x=8q\), where \(q\) is any integer. Insufficient

Combining 1 & 2: \(x=8\). Sufficient

Option C
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Re: What is the value of number X?  [#permalink]

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New post 27 Feb 2018, 22:57
2
EgmatQuantExpert wrote:

Question:



What is the value of number X?

1) The HCF of X and 36 is 4
2) The LCM of X and 36 is 72


    A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
    B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
    C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
    D) EACH statement ALONE is sufficient.
    E) Statements (1) and (2) TOGETHER are NOT sufficient.



Statement 1

Lets write the prime factorisation of 36. 36 = 2^2 * 3^2. And 4 = 2^2.
Since HCF of X and 36 is 2^2, it means X contains at least two 2's, But X cannot contain any 3's (else 3 would have also come into HCF).
Thus this tells us that X has at least 2^2 (or could have higher powers of 2), does NOT contain any 3, but might contain any other prime number than 3.

But this doesnt give us a unique value of X. Not sufficient.


Statement 2

36 = 2^2 * 3^2 and 72 = 2^3 * 3^2.
Since LCM contains product of all prime numbers with their highest powers, it means X must have three 2's in its prime factorisation (else its not possible for LCM to contain 2^3), and X CAN have only one or two 3's in its prime factorisation (because if it contained any more powers of 3, then LCM would have reflected that). And also X cannot contain any other prime number than 2 & 3 (else LCM would have reflected that).

Thus X could be either 2^3 or 2^3 * 3 or 2^3 * 3^2. Not a unique value of X. so Not Sufficient.


Combining the two statements,
X has exactly three 2's (from second statement)
X has no 3's at all (from first statement)
X does not contain any other prime number except 2 & 3 (from second statement)

Thus, X = 2^3 or 8. Sufficient.

Hence C answer
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Re: What is the value of number X?  [#permalink]

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New post 28 Feb 2018, 00:13
1
What is the value of number X?

1) The HCF of X and 36 is 4

36 when prime-factorized gives \(2^2 * 3^2\)
If the HCF of 36 and X is 4, X can be \(2^2 = 4\), \(2^3 = 8\), or \(2^2 * 5 * 7 = 140\)
Therefore, we cannot have an unique value for X (Insufficient)

2) The LCM of X and 36 is 72

36 when prime-factorized gives \(2^2 * 3^2\)
If the LCM of X and 36 is 72, X can be \(2^3 = 8\) or \(2^3 * 3 = 24\)
Therefore, we cannot have an unique value for X (Insufficient)

On combining the information given in both the statements,
using the property

HCF(x,y) * LCM(x,y) = x*y



we can arrive at a unique value of X as follows:

\(4*72 = 36*X\) -> \(X = 4*\frac{72}{36} = 8\)(Sufficient - Option C)
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Re: What is the value of number X?  [#permalink]

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New post 28 Feb 2018, 01:00

Solution:



Step 1: Analyse Statement 1:
The HCF of \(X\) and \(36\) is \(4\)
    • \(36\) can be written as:\(2^2 * 3^2\).
    • The HCF of \(X\) and \(36\) is \(4\), therefore, \(X\) should be a factor of \(4\).
      o \(X= 4k\), where \(k\) is any positive integer.
         But, can \(k\) be equal to \(3\)? Or a multiple of \(3\)?
          • No, because if \(k\) is a multiple of \(3\), it will be visible in the HCF
    o Thus, the possible values of \(X = 4,8,16,20\)….
As we do not know the exact value of \(X\),
Statement 1 alone is NOT sufficient to answer the question.
Hence, we can eliminate answer choices A and D.
Step 2: Analyse Statement 2:
The LCM of \(X\) and \(18\) is \(72\).
    • We will use the reverse process of finding the LCM and find the powers of all the prime factors in \(X\) and \(18\)
      o \(18 = 2 * 3^2\)
      o LCM \((X,18)\) = \(72\) = \(2^3 * 3^2\)
      o So, the highest power of \(2\)in the given numbers is\(3\), and the highest power of \(3\) for them is \(2\).
    However, \(18\) does not contain \(2^3\) and hence, \(X\) must contain \(2^3\) in it.
    X may or may not contain \(3/3^2\) in it, since, it is visible in the LCM.

Thus, possible values of \(X\) could be: \(8, 24, 72\)

Since we do not know the exact value of \(X\),
Statement 2 alone is NOT sufficient to answer the question.
Hence, we can eliminate answer choice B.
Step 3: Combine both Statements:
From the first Statement we got: \(X = 4,8,16,20\)… (any multiple of \(4\), but no multiples of \(12\))
From the second Statement we got: \(X = 8,24,72\)
Since \(8\) is the only number which is common in both the list, we could determine the value of \(X\)
By combining both statements we got a unique answer.
Correct Answer: Option C
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Re: What is the value of number X?  [#permalink]

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New post 28 Feb 2018, 01:40

Shortcut:



    • From the first statement, we will not get any unique value of X. Possible values of \(X = 4, 8 , 16\) etc.
    • From the second statement, we will not get any unique value of X. Possible values of \(X = 8, 24, 72\).
    • But we know that Product of two numbers = LCM * GCD of both the numbers.
      o Hence, \(36*X\) = \(72*4\)
      o Or, \(X = 8\).

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Re: What is the value of number X?  [#permalink]

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Re: What is the value of number X?   [#permalink] 20 May 2019, 01:10
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