Is y!/x! an integer?

With (1) we know that x^2 - y^2 = 5! +1

Is y!/x! an integer?

Well, that's a factorial divided by a factorial. We know that 0! = 1.
And since x^2 - y^2 = some positive integer, we know that x and y are going to be integers as well (no decimals, a 7.13! would be weird!)

So...x and y are integers. y! / x! ...okay we well can get into dangerous territory if the denominator is a large number --- that introduces fractions. If y>x, then numerator will always be larger and we won't necessarily get that fraction problem.
But in this case, we know that x^2 - y^2 = some positive integer...

... so x>y. That means we know DEFINITIVELY the denominator is going to be bigger than the numerator and we know DEFINITIVELY that we get a fraction less than one.

In that case, is y! / x! an integer? Well we know definitively that it's a fraction less than 1, thus not an integer. Thus, we have enough information to say whether YES integer or NO, not an integer. The answer would be NO not an integer. But don't confuse that with the actual question it's asking...it's not do we have an integer yes or no? THe question is asking...do we have SUFFICIENT INFORMATION to determine whether the answer to that question is yes or no. And the answer is we DO have SUFFICIENT INFORMATION to make a definitive yes or no answer.

1) (x+y)(x-y)=5!+1
121*1=121(no other combination is possible. 121=11*11 or 121*1
Since sum and difference cannot be same.hence we are left with only 121*1)
This means x=61, y=60 i.e y!/x! Is not an integer. Hence sufficient.
2) x+y = 121
This can have multiple values for x & y. Hence not sufficient.

Sent from my ONEPLUS A3003 using GMAT Club Forum mobile app

Hi,

please post the Q along with topic name and OA.
y and x are integers

Nw for the Q..
when can \(\frac{y!}{x!}\) be an integer.. when either y>x or y=x..

so lets check the statements
(1) (x + y)(x − y) = 5! + 1
as we are looking for y!/x!, we can take that y and x are positive integers as negative / fractions do not have factorials..
RHS is positive and LHS has x-y, so x-y must be POSITIVE and hence x>y..
so our answer will be NO always.
y!/x! will be a fraction.
suff

(2) x + y = 11^2
so x+y =121..
x and y can take various values.
with y as 61 and x as 60 ans is YES
with y as 1 and x as 120 ans is NO
insuff

A

[quote="ashudhall"]1) (x+y)(x-y)=5!+1
121*1=121(no other combination is possible. 121=11*11 or 121*1
Since sum and difference cannot be same.hence we are left with only 121*1)
This means x=61, y=60 i.e y!/x! Is not an integer. Hence sufficient.
2) x+y = 121
This can have multiple values for x & y. Hence not sufficient.


Hi,
Thanks for the reply!
But for the 1st statement, if x+y =11 and x-y =11, it leaves us with x=11 and y=0.
Thus making the of (y!/x!) as 0.
And the other case as you said, not an integer.
So, shouldn't the answer be C?

Hi,
Thanks for the reply.
My query is for statement (1), there is a possibility that x=11 and y=0, or x=121 and y=1.
(Statement 2 is insufficient alone.)
So this is where statement 2 helps us. So, shouldn't the answer be C?

Sent from my ONE A2003 using GMAT Club Forum mobile app

0! is 1 not 0, hence in that case as well 1/11! Would not be an integer. Hence the answer should be A only.

Sent from my ONEPLUS A3003 using GMAT Club Forum mobile app

Oh God. Missed that! Thanks

Sent from my ONE A2003 using GMAT Club Forum mobile app

This question is basically testing whether x! Completely divides into y! Or not. In other words, what you have to check is whether X! < Y! or X! > Y! as Factorial is basically a product of all numbers from n to 1.

For e.g. 5! = 5*4*3*2*1


S1 -> (X+Y)(X-Y) =5! + 1
(X+Y)(X-Y) = 121
X=61 , Y=60 => Y!/X! Is not an integer.
X = 60, Y = 61 violates the equation as the left hand side becomes negative => -121 is not equal to 121.
Sufficient.

S2 -> X+Y = 121
Values can interchange I.e. X =60 and Y=61 and vice versa. Insufficient.

A is the answer.

Hope this helps.