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# If k is a multiple of 24 but not a multiple of 16, which of the follow

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If k is a multiple of 24 but not a multiple of 16, which of the follow  [#permalink]

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13 Aug 2018, 04:54
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If k is a multiple of 24 but not a multiple of 16, which of the following cannot be an integer?

(A) $$\frac{k}{8}$$

(B) $$\frac{k}{9}$$

(C) $$\frac{k}{32}$$

(D) $$\frac{k}{36}$$

(E) $$\frac{k}{81}$$

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Re: If k is a multiple of 24 but not a multiple of 16, which of the follow  [#permalink]

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13 Aug 2018, 08:07
Bunuel wrote:
If k is a multiple of 24 but not a multiple of 16, which of the following cannot be an integer?

(A) $$\frac{k}{8}$$

(B) $$\frac{k}{9}$$

(C) $$\frac{k}{32}$$

(D) $$\frac{k}{36}$$

(E) $$\frac{k}{81}$$

k is a multiple of 24 but not a multiple of 16.
k=2x2x2x3xp
k is not equal to 2x2x2x2xq
A) Integer when k=24
B) Integer when k=72
C) Not an integer since each multiple of 32 will definitely be a multiple of 16
D) Integer when k=72
E) Integer when $$k=2X2X2X3^4$$

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If k is a multiple of 24 but not a multiple of 16, which of the follow  [#permalink]

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13 Aug 2018, 20:43
1
Bunuel wrote:
If k is a multiple of 24 but not a multiple of 16, which of the following cannot be an integer?

(A) $$\frac{k}{8}$$

(B) $$\frac{k}{9}$$

(C) $$\frac{k}{32}$$

(D) $$\frac{k}{36}$$

(E) $$\frac{k}{81}$$

We have $$k=2^3*3^1*$$(any odd integer)Let's check the options:-
A. k/8 is an integer when k=24*any odd integer
B. k/9 is an integer when k=24*3*any odd integer
C. $$\frac{k}{32}$$ can never be an integer . ($$32=2^5$$, we can't make the odd integer=$$2^2$$ since k is not a multiple of 16)
D. $$36=2^2*3^2=2^3*3^1*\frac{3}{2}$$ , so k/36 is an integer.
E. 81=3^4=2^3*3^1*\frac{3^3}{2^3}[/m], so k/36 is an integer.

Ans. (C)
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Re: If k is a multiple of 24 but not a multiple of 16, which of the follow  [#permalink]

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18 Aug 2018, 19:29
Bunuel wrote:
If k is a multiple of 24 but not a multiple of 16, which of the following cannot be an integer?

(A) $$\frac{k}{8}$$

(B) $$\frac{k}{9}$$

(C) $$\frac{k}{32}$$

(D) $$\frac{k}{36}$$

(E) $$\frac{k}{81}$$

If k is a multiple of 24, then k could be:

24, 48, 72, 96, 120, 144, …

However, since k is not a multiple of 16, then k can’t be:

48, 96, 144, …

So we see that k could only be:

24, 72, 120, …

That is, k is an odd multiple of 24.

We see that k/8 is an integer since 24 is divisible by 8. Furthermore, k/9 and k/81 could each be an integer if k is 9 x 24 and 81 x 24, respectively. Lastly, k/36 could be an integer if k is 9 x 24 (notice that 9 x 24 = 9 x 4 x 6 = 36 x 6). Therefore, k/32 can’t be integer (notice that 32 = 16 x 2, so if k is not a multiple of 16, then k can’t be a multiple of 32).

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Re: If k is a multiple of 24 but not a multiple of 16, which of the follow  [#permalink]

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19 Aug 2018, 06:46

Solution

Given:
• k is a multiple of 24 but not a multiple of 16

To find:
• The option which cannot be an integer.

Approach and Working:
• K is multiple of 24 then K can be written as $$2^{(3+a)}*3^{(1+b)}* some other prime factors$$.
o However, k is not a multiple of 16.
o So, a must be 0.
• Thus, K= $$2^3+*3^{(1+b)}* some other prime factors$$.

Now, among the given options, only $$\frac{K}{32}$$ will not be an integer.

Hence, the correct answer is option C.

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Re: If k is a multiple of 24 but not a multiple of 16, which of the follow  [#permalink]

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19 Aug 2018, 07:19
Bunuel wrote:
If k is a multiple of 24 but not a multiple of 16, which of the following cannot be an integer?

(A) $$\frac{k}{8}$$

(B) $$\frac{k}{9}$$

(C) $$\frac{k}{32}$$

(D) $$\frac{k}{36}$$

(E) $$\frac{k}{81}$$

We are given: k is a multiple of 24, k can be any of the following: 24, 48, 72, 96, 120, 144...

We are also told: k is not a multiple of 16. Thus, we can eliminate: 48, 96, 144...

We are left with values: 24, 72, 120...

On substituting in the values above in the given options, 24/32 , 72/32, 120/32 are not integers.

Thus, the answer is C.
Re: If k is a multiple of 24 but not a multiple of 16, which of the follow &nbs [#permalink] 19 Aug 2018, 07:19
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