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For any integer n greater than 1, factorial denotes the

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For any integer n greater than 1, factorial denotes the

by swerve » Mon Sep 17, 2018 3:31 pm

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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 aa and bb 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.

The OA is A

Source: e-GMAT
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by deloitte247 » Sun Sep 23, 2018 9:30 am

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We are given two positive integers aa and bb such that b > a
Question = Find the total number of factors of the target number which divides the factorials of both a and b since factorials is the product of all integers from 1 to n inclusive.
* Factor of b would consist of product of all the numbers from 1 to b.
* Factors of a would consist of product of all the number from 1 to a.
* b > a means that factorial of b would consist of all the numbers present in the factorial of a.
* The largest number which divides the factorial of both a and b and this would be the factorial of a itself.
* If we can calculate the value of a, we would get the answer.

STATEMENT 1 =
a is the greatest integer for which,
$$3^a\ is\ a\ factor\ of\ product\ of\ integers\ from\ 1\ to\ 20\ inclusive.$$
$$This\ statement\ means\ that\ a\ =\ \frac{20!}{3^a}$$
For finding a, we have to calculate how many 3 are there in 20!. We can do that by dividing 20 with 3 in the first and second exponent and add the result.
I. e $$a\ =\ \frac{20}{3!}=\frac{20}{3}=\ 6\ $$
$$a\ =\ \frac{20}{3^2}=\frac{20}{9}=\ 2\ $$
a = 6 + 2 = 8
Statement 1 is SUFFICIENT.

Statement 2 = b is the largest possible number that divides positive integer n,
$$where\ n^3\ is\ divisible\ by\ 96.$$
This statement does not provide sufficient information about a, hence statement 2 is INSUFFICIENT.
Option A IS CORRECT.
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