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For two positive integers A & B, what is the highest num

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by RBBmba@2014 » Tue May 12, 2015 3:03 am
For two positive integers A & B, what is the highest number that divides completely the product of integers from 1 to A and 1 to B such that B = A + 29.

(A) 1
(B) Product of all integers from 1 to A
(C) Product of all integers from 1 to B
(D) 29*A
(E) Can't be determined

OA: B

P.S: Interesting question. Although I got this,I'd be looking forward to some better explanation from Experts.
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by theCEO » Tue May 12, 2015 3:41 am
RBBmba@2014 wrote:For two positive integers A & B, what is the highest number that divides completely the product of integers from 1 to A and 1 to B such that B = A + 29.

(A) 1
(B) Product of all integers from 1 to A
(C) Product of all integers from 1 to B
(D) 29*A
(E) Can't be determined

OA: B

P.S: Interesting question. Although I got this,I'd be looking forward to some better explanation from Experts.
B = A + 29
From this we know that B is greater than A
Therefore B! is greater than A!
B! can be rewritten as i x A! where i is a constant

So the question reads what is the highest number that can divide A! and iA!
A!/A! = 1
iA!/A!= i

Therefore the answer is A! = B
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by GMATGuruNY » Tue May 12, 2015 5:00 am
RBBmba@2014 wrote:For two positive integers A & B, what is the highest number that divides completely the product of integers from 1 to A and 1 to B such that B = A + 29.

(A) 1
(B) Product of all integers from 1 to A
(C) Product of all integers from 1 to B
(D) 29*A
(E) Can't be determined
The product of the integers from 1 to A = A!.
Since B=A+29, the product of the integers from 1 to B = B! = (A+29)!.

Question stem, rephrased:
What is the greatest common factor of A! and (A+29)!?

Case 1: A=1
Here, A! = 1! and (A+29)! = 30!.
The GCF of 1! and 30! = 1!.

Case 2: A=10
Here, A! = 10! and (A+29)! = 39!.
The GCF of 10! and 39! = 10!.

Case 3: A=100
Here, A! = 100! and (A+29)! = 129!.
The GCF of 100! and 129! = 100!.

In every case, the GCF = A!.

The correct answer is B.
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by RBBmba@2014 » Tue May 12, 2015 5:10 am
@ GMATGuru - I did it in the same way (except plugging in numbers), so I believe this is the BEST way to solve this sort of problems! Thanks.
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by Brent@GMATPrepNow » Tue May 12, 2015 7:17 am
RBBmba@2014 wrote:For two positive integers A & B, what is the highest number that divides completely the product of integers from 1 to A and 1 to B such that B = A + 29.

(A) 1
(B) Product of all integers from 1 to A
(C) Product of all integers from 1 to B
(D) 29*A
(E) Can't be determined
NOTE: The question is asking us to find the greatest common factor of A! and B!

Given: B = A + 29

Product of integers from 1 to A = (1)(2)(3)...(A)
Product of integers from 1 to B = (1)(2)(3)...(A)(A+1)(A+2)...(A+28)(A+29)

As you can see, both products SHARE the product (1)(2)(3)...(A), and they share nothing more.
So, (1)(2)(3)...(A) must be the greatest common factor of both products.

Answer: B

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by prachi18oct » Wed May 13, 2015 9:56 am
Hello,

I interpreted the question as below :
What is the highest number that divides completely the product of (1 to A) and (1 to B)
I took it in other way :-
highest number that divides completely => (1.2.3.....A)(1.2.3......A+29)
SO it should have been A(A+29).
I think the language of the question is not very clear, atleast to me.
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