If A and B are positive integers and 24AB is a perfect square, then

Question

If A and B are positive integers and 24AB is a perfect square, then which of the following cannot be possible?

I. Both A and B are odd.
II. AB is a perfect square
III. Both A and B are divisible by 6

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I, II, and III

sanjitscorps18 · Answer

24ab is a perfect square.

Hence ab = 6 x k^2

1. If both and B are odd then 2 would never have a pair in the square hence not possible.

2. ab cannot be a perfect square as 6 would not have a pair and only k^2 would have a proper root.

3. If a and b both are divisible by 6 then there is a possibility that 24ab may or may not be a perfect square. Hence we can keep this.

Hence Option D should be the answer.

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parasma · Answer

2401 is the only perfect square in 2400-2499.

So AB=01

ONLY b is true.
SO answer is D.

PgRaul · Answer

If A and B are positive integers and 24 x A x B is a perfect square, which of the following CANNOT be possible?
I. Both A and B are odd. II. AB is a perfect square. III. Both A and B are divisible by 6. 
I think it is 24*A*B and not the 4 digit no 24AB

Crytiocanalyst · Answer

I. Both A and B are odd.
we need 2 as a factor in order to arrive at square therefore not possible

II. AB is a perfect square
let us assume it to be 36 then it doesn't become perfect square any possibility from fom 16 to 81 not possible

III. Both A and B are divisible by 6

Therefore IMO D

Bunuel · Answer

This question is a part of  Are You Up For the Challenge: 700 Level Questions  collection.

Pranavsawant · Answer

Alright look. 2401 is the only perfect square in the range 2400-2499
So A = 0, B = 1 is the only possible soln. However they say A, B are both positive integers. 0 is not positive. So we have no solns.

1. Both A and B are odd. If both A and B are odd, 24AB is not a perfect square. So it cannot be possible 
2) If AB is a perfect square then this would have been possible if we allow 0 as a value. But we don't so again, not possible
3) Not possible again as only soln is A = 0, B = 1

All 3 statements are not possible. In fact, no matter what you write in any statements, it won't make the truth of the original proposition. 24AB can be a perfect square if and only if A = 0, B =1. But 0 is not in our domain of values. All 3 statements cannot be true regardless.

If anything, this is a bad bad question since the original proposition (24AB being a perfect square) can  never  be true for positive integers A, B

Bunuel  please correct the answer if i'm right

gaa · Answer

Not Bunuel but I'm confident that the question means 24AB as in 24 * A * B, not a 4-digit number with digits 2, 4, A, and B. Interpreting it that way makes both statements 1) and 2) impossible.

Bunuel · Answer

Yes, 24AB means 24 * A * B. If 24AB were a four-digit number, it would have been explicitly mentioned. Without such clarification, "24AB" can only represent "24 * A * B", as the multiplication sign (*) is typically omitted.

Kinshook · Answer

Bunuel  answer choice D should be 
(D) I and III only
Please correct

If A and B are positive integers and 24AB is a perfect square, then which of the following cannot be possible?

I. Both A and B are odd.
II. AB is a perfect square
III. Both A and B are divisible by 6

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III

2401 is the only perfect square of the form 24AB. 
Therefore, AB = 01 which is a perfect square, A is even, B is NOT divisible by 6.

IMO D

Pranavsawant · Answer

Here 24AB means 2*4*A*B apparently lol 
Even I made the same reasoning mistake above.

Pranavsawant · Answer

Interesting but then I don't get why they would put it as 24AB 
They should've just said it as 8*A*B 
24AB is such a misleading way to represent 8*A*B

But yeah D is obvious in that case. Would never expect an OG question to trip us like this

If A and B are positive integers and 24AB is a perfect square, then

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If A and B are positive integers and 24AB is a perfect square, then

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards