For any integer n greater than 1, factorial denotes the product of all

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