If t is divisible by 12, what is the least possible integer value of

Question

If t is divisible by 12, what is the least possible integer value of a for which t^2/2^a might not be an integer?

(A) 2
(B) 3
(C) 5
(D) 6
(E) 40

IanStewart · Answer

Since we're dividing by 2^a, we only care about 2's here, so when we read "t is divisible by 12", we don't care that t is divisible by 3. We care that t is divisible by 2^2. That means t^2 is divisible by (2^2)^2 = 2^4. So we know if we divide t^2 by any power of 2 up to 2^4, we'll get an integer. But if we divide by 2^5, we are not guaranteed to get an integer. So the answer is 5.

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Showmeyaa · Answer

If  \frac{t}{12} = integer , minimum possible value of t as an integer = 12
Similarly, minimum possible value of  t^2 = 144 = 2^4 * 3^2 
So, least possible value for  2^a = 5  for it to make a non-integer. IMO, (C)!

Kinshook · Answer

Asked: If t is divisible by 12, what is the least possible integer value of a for which t^2/2^a might not be an integer?

Let t = 12k ; where k is an integer

t^2/2^a = (12k)^2/2^a = 144k^2/2^a 
Since 144 = 2^4*3^2
If k is an odd number; t^2 is divisible by 2^4 but may not be divisible by 2^5; t^2/2^5 may not be an integer

IMO C

BrushMyQuant · Answer

Given that t is divisible by 12 and we need to find what is the least possible integer value of a for which  \frac{t^2}{2^a}  might not be an integer

t is divisible by 12 => t is a multiple of 12 => t = 12x where x is an integer

\frac{t^2}{2^a}  =  \frac{(12x)^2}{2^a}  =  \frac{(2^2*3*x)^2}{2^a}  =  \frac{2^4*3^2*x^2}{2^a}

Now, in numerator 2 has a power of 4. So, a should be more than 4 for  \frac{t^2}{2^a}  to not be an integer
=> a = 5

So,  Answer will be C .
Hope it helps!

avigutman · Answer

If t is divisible by 12, it contains at least two factors of 2. Therefore, t^2 contains at least four factors of 2.
t^2/2^a will definitely be an integer as long as a is no more than 4. If a exceeds 4, there's a chance that t^2/2^a isn't an integer. The least possible integer exceeding 4 is 5. Answer C.

AlexTheTrainer · Answer

Premise: t/(2^2 * 3) ——> must be Integer

Corollary:  ^2 ——> must be Integer

So: t^2/(2^4 * 3^2) ——> must be Integer

Thus: t^2/(2^a) ——> must be Integer WHEN a < 5.

Answer C

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If t is divisible by 12, what is the least possible integer value of

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If t is divisible by 12, what is the least possible integer value of

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