For any integer \(n\) greater than \(1,\) factorial denotes the product of all the integers from \(1\) to \(n,\) inclusi

[Vincen]

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 the 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 \(96.\)

Answer: A

Source: e-GMAT

[swerve]

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 the 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 \(96.\)

Answer: A

Source: e-GMAT

1) We are asked about how many 3's are there in 20! if we factor it: we can answer that question easily
\(1 \cdot 2 \cdot 3(1) \cdot 4 \cdot 5 \cdot 6(1) \cdot 7 \cdot 8 \cdot 9(2)\cdot10\cdot 11\cdot 12(1)\cdot ... \cdot 15(1)\cdot ... \cdot 18(2): 1 + 1 + 2 + 1 + 1 =2 =8,\) so \(a = 8\) and so we can answer our question. Sufficient \(\Large{\color{green}\checkmark}\)

2) \(b\) is \(n\) and \(n\) can be as big as one wishes it to be. In this case we don't have any idea about \(a\) except for the fact that it's lower than \(b,\) but yet again, \(a\) can be anything in this case, thus our answer is unknown. Not Sufficient. \(\Large{\color{red}\chi}\)

Therefore, A