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Updated on: 10 Mar 2024, 10:33
Originally posted by EgmatQuantExpert on 09 Apr 2015, 02:17.
Last edited by Bunuel on 10 Mar 2024, 10:33, edited 7 times in total.
Last edited by Bunuel on 10 Mar 2024, 10:33, edited 7 times in total.
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A
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For any integer \(n\) greater than 1, factorial denotes the product of all the integers from 1 to \(n\), inclusive. It’s given that \(a\) and \(b\) are two positive integers such that \(b > a\). What is the total number of factors of the largest number that divides the factorials of both \(a\) and \(b\)?
(1) \(a\) is the greatest integer for which \(3^a\) is a factor of product of integers from 1 to 20, inclusive.
(2) \(b\) is the largest possible number that divides positive integer \(n\), where \(n^3\) is divisible by 96.
(1) \(a\) is the greatest integer for which \(3^a\) is a factor of product of integers from 1 to 20, inclusive.
(2) \(b\) is the largest possible number that divides positive integer \(n\), where \(n^3\) is divisible by 96.
This is
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Updated on: 07 Aug 2018, 04:42
Originally posted by EgmatQuantExpert on 10 Apr 2015, 07:01.
Last edited by EgmatQuantExpert on 07 Aug 2018, 04:42, edited 1 time in total.
Last edited by EgmatQuantExpert on 07 Aug 2018, 04:42, edited 1 time in total.
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Detailed Solution
Step-I: Given Info:
We are given two positive integers \(a\) and \(b\) such that \(b > a\). We are asked to find the total number of factors of the largest number which divides the factorials of both \(a\) and \(b\).
Step-II: Interpreting the Question Statement
Since factorial is the product of all integers from 1 to \(n\) inclusive:
i. factorial of \(b\) would consist of product of all the numbers from 1 to \(b\)
ii. factorial of \(a\) would consist of product of all the numbers from 1 to \(a\)
As \(b > a\), this would imply that factorial of \(b\) would consist of all the numbers present in factorial of \(a\). For example factorial of 30 would consist of all the numbers present in factorial of 20.
So, the largest number which divides the factorial of both \(b\) and \(a\), i.e. the GCD of factorial of \(b\) and \(a\) would be the factorial of \(a\) itself. So, if we can calculate the value of \(a\), we would get to our answer.
Step-III: Statement-I
Statement-I tells us that \(a\) is the greatest integer for which \(3^a\) is a factor of factorial of 20. Since we can calculate the number of times 3 comes as a factor of numbers between 1 to 20, we can find the value of \(a\).
Thus Statement-I is sufficient to answer the question.
Please note that we do not need to actually calculate the value of \(a\). Just the knowledge, that we can calculate the unique value of \(a\) is sufficient for us to get to our answer.
Step-IV: Statement-II
Statement-II tells us that \(b\) is the largest possible number that divides \(n\), where \(n^3\) is divisible by 96.
Note here that the statement talks only about \(b\) and nothing about \(a\). Since, we do not have any relation between \(b\) and \(a\) which would give us the value of \(a\), if we find \(b\), we can say with certainty that this statement is insufficient to answer the question.
Again, note here that we did not solve the statement as we could infer that it’s not going to give us the value of \(a\), which is our requirement.
Step-V: Combining Statements I & II
Since, we have received our unique answer from Statement-I, we don’t need to combine the inferences from Statement-I & II.
Hence, the correct answer is Option A
Key Takeaways
1. Familiarize yourself with all the names by which the test makers can call the GCD and the LCM.
For example,
• GCD is also known as the HCF
• GCD can also be described as ‘the largest number which divides all the numbers of a set’
• LCM of a set of numbers can also be described as ‘the lowest number that has all the numbers of that set as it factors’
2. Since factorial is product of a set of positive integers, the GCD of a set of factorials would always be the factorial of the smallest number in the set
Zhenek- Brilliant work!!, except that we did not need the calculation in St-I
Harley1980- Kudos for the right answer, two suggestions- calculation not needed in St-I and in St-II you calculated the least possible value of \(b\), which was again not needed as it did not tell us anything about \(a\).
Regards
Harsh
Step-I: Given Info:
We are given two positive integers \(a\) and \(b\) such that \(b > a\). We are asked to find the total number of factors of the largest number which divides the factorials of both \(a\) and \(b\).
Step-II: Interpreting the Question Statement
Since factorial is the product of all integers from 1 to \(n\) inclusive:
i. factorial of \(b\) would consist of product of all the numbers from 1 to \(b\)
ii. factorial of \(a\) would consist of product of all the numbers from 1 to \(a\)
As \(b > a\), this would imply that factorial of \(b\) would consist of all the numbers present in factorial of \(a\). For example factorial of 30 would consist of all the numbers present in factorial of 20.
So, the largest number which divides the factorial of both \(b\) and \(a\), i.e. the GCD of factorial of \(b\) and \(a\) would be the factorial of \(a\) itself. So, if we can calculate the value of \(a\), we would get to our answer.
Step-III: Statement-I
Statement-I tells us that \(a\) is the greatest integer for which \(3^a\) is a factor of factorial of 20. Since we can calculate the number of times 3 comes as a factor of numbers between 1 to 20, we can find the value of \(a\).
Thus Statement-I is sufficient to answer the question.
Please note that we do not need to actually calculate the value of \(a\). Just the knowledge, that we can calculate the unique value of \(a\) is sufficient for us to get to our answer.
Step-IV: Statement-II
Statement-II tells us that \(b\) is the largest possible number that divides \(n\), where \(n^3\) is divisible by 96.
Note here that the statement talks only about \(b\) and nothing about \(a\). Since, we do not have any relation between \(b\) and \(a\) which would give us the value of \(a\), if we find \(b\), we can say with certainty that this statement is insufficient to answer the question.
Again, note here that we did not solve the statement as we could infer that it’s not going to give us the value of \(a\), which is our requirement.
Step-V: Combining Statements I & II
Since, we have received our unique answer from Statement-I, we don’t need to combine the inferences from Statement-I & II.
Hence, the correct answer is Option A
Key Takeaways
1. Familiarize yourself with all the names by which the test makers can call the GCD and the LCM.
For example,
• GCD is also known as the HCF
• GCD can also be described as ‘the largest number which divides all the numbers of a set’
• LCM of a set of numbers can also be described as ‘the lowest number that has all the numbers of that set as it factors’
2. Since factorial is product of a set of positive integers, the GCD of a set of factorials would always be the factorial of the smallest number in the set
Zhenek- Brilliant work!!, except that we did not need the calculation in St-I
Harley1980- Kudos for the right answer, two suggestions- calculation not needed in St-I and in St-II you calculated the least possible value of \(b\), which was again not needed as it did not tell us anything about \(a\).
Regards
Harsh
09 Apr 2015, 02:49
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EgmatQuantExpert
1) we know that \(20!/3^a\). For finding a we should calculate 3 in 20!
We can use a shortcut by dividing 20 on the 3 in 1 power, than in second power etc. and sum the results and this will be A
20 / 3 = 6; 20 / 9 = 2; 6 + 2 = 8 so 20! can be divided on \(3^8\) and A = 8
And at first glance we can think that this isn't enough because we don't know B
But question asks about number of factors of biggest number that can divide both A! and B!
So A! will be divide B! because B bigger than A
So now we should only calculate number of factors 8! We don't need to do this on the exam because we know that we can do it.
But method is: find all prime factors of number, take their powers add to each power 1 and sum these numbers, this will be number of factors
1*2*3*4*5*6*7*8
\(2^7 * 3^2 * 5^1 * 7^1\)
sum of powers + 1: (7+1)*(2+1)*(1+1)*(1+1) = 8*3*2*2 = 96
Sufficient
2) This statement insufficient because we don't know about A and for deciding task we should know about A
How we can find B?
For find B we should find N and we know that N^3 is divisible by 96
\(96 = 2^5*3\)
So to be divisible by 96, \(N^3\) should have prime factors 2 and 3 and this factors should be in powers multiple of 3
And least possible number will be \(2^6*3^3\) and this will be equal \(12^3\)
So the least possible B equal to 12 but we don't know about A so this fact Insufficient
And answer is A
