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Updated on: 10 Oct 2022, 06:54
Originally posted by SaquibHGMATWhiz on 06 Oct 2022, 07:13.
Last edited by SaquibHGMATWhiz on 10 Oct 2022, 06:54, edited 4 times in total.
Last edited by SaquibHGMATWhiz on 10 Oct 2022, 06:54, edited 4 times in total.
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
(hard)
Question Stats:
49% (01:51) correct
51%
(01:46)
wrong
based on 121
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The number \(N = 2^3 × 3^4 × p^2\), where p is a prime number. How many even factors does the number N have?
1. N has more than 30 factors.
2. N has fewer than 70 factors.
1. N has more than 30 factors.
2. N has fewer than 70 factors.
Updated on: 10 Oct 2022, 07:14
Originally posted by SaquibHGMATWhiz on 06 Oct 2022, 07:35.
Last edited by SaquibHGMATWhiz on 10 Oct 2022, 07:14, edited 2 times in total.
Last edited by SaquibHGMATWhiz on 10 Oct 2022, 07:14, edited 2 times in total.
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This is our solution to this question
Solution:
Step 1: Analyse Question Stem:
We are given a number \(N=2^3×3^4×p^2\), where p is a prime number
Case 1: p = 2
Case 2: p = 3
Case 3: p = any prime greater than 3
Step 2: Analyze Statements Independently (And Eliminate Options) - AD/BCE
Statement 1: N has more than 30 factors
Statement 2: N has fewer than 70 factors.
The answer is option A.
Note: We have made a small change to the second statement of the question.
This question is created to test the knowledge related to the number of factors of a number. We have published an article can on the same. You can view it using this LINK
Solution:
Step 1: Analyse Question Stem:
We are given a number \(N=2^3×3^4×p^2\), where p is a prime number
- Here p can be 2, 3, or any other prime greater than 3. Let us take each case.
Case 1: p = 2
- \(N=2^5×3^4\)
Total factors =\((5+1)×(4+1)=6×5=30 \)
Case 2: p = 3
- \(N=2^3×3^6\)
Total factors =\((3+1)×(6+1)=4×7=28 \)
Case 3: p = any prime greater than 3
- \(N=2^3×3^4×p^2 \)
Total factors \(=(3+1)×(4+1)×(2+1)=4×5×3=60 \)
Step 2: Analyze Statements Independently (And Eliminate Options) - AD/BCE
Statement 1: N has more than 30 factors
- Information in this statement matches with case 3 of our pre-analysis
This means p is a prime number greater than 3 and\( N=2^3×3^4×p^2\)
Now we can very easily find its number of odd factors and even factors using respective formulas
Thus, statement 1 alone is sufficient and we can eliminate options B, C, and E
Statement 2: N has fewer than 70 factors.
- Information in this statement matches with all the cases of our pre-analysis
\(N=2^5×3^4\), \(N=2^3×3^6\), and \(N=2^3 × 3^4 × p^2\) will have different numbers of even factors
Thus, statement 2 alone is not sufficient
Thus, we can eliminate option D.
The answer is option A.
Note: We have made a small change to the second statement of the question.
This question is created to test the knowledge related to the number of factors of a number. We have published an article can on the same. You can view it using this LINK
09 Oct 2022, 21:55
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SaquibHGMATWhiz
The # of even factors will depend on the even/odd status of p, and also the total # of factors.
1. N has more than \(30\) factors.
IF p is even i.e. \(2\) Then we have \(2^5 *3^4\) and total factors \(= (5+1)(4+1) = 30\)
IF p is odd i.e. \(3 \) Then we have \(2^3 *3^6 \) and total factors \(= (3+1)(6+1) = 28\)
Hence we know p is odd prime \(>3 \)
So total factors become \((3+1 )(4+1)(2+1) = 60\) and even factors \(60 -15 = 45\)
SUFF.
2. N has fewer than 50 factors.
IF p is even i.e. \(2\) Then we have \(2^5 *3^4\) and total factors \(= (5+1)(4+1) = 30\)
IF p is odd i.e. \(3\) Then we have \(2^3 *3^6\) and total factors \(= (3+1)(6+1) = 28\)
Hence all we can say is that p could be \(2\) or \(3\) leading to different answers .
EDIT: After the second statement has been altered from the original version , we can say p could also be any other prime greater than \(3\), leading to different answers.
INSUFF.
Ans A
Hope it's clear.
