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

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Re: If t is divisible by 12, what is the least possible integer value of a [#permalink]
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Bunuel wrote:
If t is divisible by 12, what is the least possible integer value of a for which $$\frac{t^2}{2^a}$$ might not be an integer?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

t is divisible by 12. It means that t is a multiple of 12.

Let's assume t is 12.

$$\frac{12^2}{2^a}$$

=$$\frac{3^2 * 2^4}{2^a}$$

= $$3^2 * 2^{(4 - a)}$$

= $$9 * 2^{(4-5)}$$

= $$9 * 2^{(-1)}$$

For negative exponents we get fraction.

So, least possible integer value of a is 5.

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If t is divisible by 12, what is the least possible integer value of a [#permalink]
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Bunuel wrote:
If t is divisible by 12, what is the least possible integer value of a for which $$\frac{t^2}{2^a}$$ might not be an integer?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

$$12^2=(3*2^2)^2$$=$$3^2*2^4$$

Now, $$\frac{t^2}{2^a}$$=$$\frac{3^2*2^4}{2^a}$$

The given fraction is an integer when a=2,3,4, and 6 among options.

Ans. (D)
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Re: If t is divisible by 12, what is the least possible integer value of a [#permalink]
Bunuel wrote:
If t is divisible by 12, what is the least possible integer value of a for which $$\frac{t^2}{2^a}$$ might not be an integer?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Since the smallest positive value of t is 12, and since 12 = 2^2 x 3^1, we see that 12^2 = 2^4 x 3^2. This factorization shows us that any power of 2, from 2^1 to 2^4, inclusive, is guaranteed to divide into t^2. Thus, we are sure that when a is 1, 2, 3, or 4, t^2/2^a is an integer.

Therefore, the smallest value of a such that t^2/2^a might not be an integer is 5.

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Re: If t is divisible by 12, what is the least possible integer value of a [#permalink]
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Hi, Lets assume t=48 and if we try to solve it, it is divisible by 2^5. Can you please explain why this is happening?
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Re: If t is divisible by 12, what is the least possible integer value of a [#permalink]
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akankshaboparai wrote:
Hi, Lets assume t=48 and if we try to solve it, it is divisible by 2^5. Can you please explain why this is happening?

Hi akankshaboparai,
Question stem:- what is the least possible integer value of a for which $$\frac{t^2}{2^a}$$ MIGHT NOT be an integer?

1) We have been asked to find the least possible integer value of 'a'.
2) As assumed, at t=48 and a=5, the given fraction is an integer. (Correct, but least possible value of 'a' would be obtained when 't' is the minimum. ). Does a=5 produce integer values of the fraction at all 't'? NO (Discard this scenario)
3) Here t=12k, k is a positive integer.
4) So we have to check at t=12*1=12 (At minimum 't')
$$\frac{12^2}{2^a}$$=$$\frac{2^4*3^2}{2^a}$$ doesn't yield an integer at a=5 and 6. But it yields integers at a=1,2,3, and 4.

Between 5 and 6, 5 is the least possible value.

Hope it helps.
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Re: If t is divisible by 12, what is the least possible integer value of a [#permalink]
A simple step, but seemingly skipped by others. We can't use t=0 as 0 can be divided by any other number (except 0) resulting in an integer.
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Re: If t is divisible by 12, what is the least possible integer value of a [#permalink]
Bunuel wrote:
If t is divisible by 12, what is the least possible integer value of a for which $$\frac{t^2}{2^a}$$ might not be an integer?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

t=12k

So question: (k^2 * 9 * 2^4) / 2^a, if a=5 then the expression might not be an integer.
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Re: If t is divisible by 12, what is the least possible integer value of a [#permalink]
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Re: If t is divisible by 12, what is the least possible integer value of a [#permalink]
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