If x is a positive integer, is x a multiple of 12?

Question

(1) x^2 is a multiple of 96
(2) x^2 is a multiple of 24

Shiwangvictory123 · Answer

case 1: 96=2^5 * 3
X^2= 2^5 * 3
X=2^3 * 3
which is a multiple of 12
case 2: 24=2^3 * 3
X^2=2^3 * 3
X=2^2 * 3
which is a multiple of 12
Answer is D

deepalidmoon · Answer

Can you explain the concept behind why X= 2^3*3 for X^2=2^5*3. I am missing this part.

deepalidmoon · Answer

Bunuel : Could you explain the concept behind why X= 2^3*3 for X^2=2^5*3.

deepalidmoon · Answer

bb  can you please help me out with this problem?

SHoylandBPrep · Answer

Stat (1) 
This is a tricky problem. You need to think about prime factorization. Find the P.F. of 96.

>  96 = 2^5 × 3^1

When you square something, the exponents on the prime factors are multiplied by 2, so they will be even. Thus, the exponents of  x^2  must be even. You need to bump the exponents up to the next even number.

>  x^2  is a multiple of  2^6 × 3^2

>  x  is a multiple of  2^3 × 3^1 = 24

In other words, since  x^2  must be a multiple of 96,  x  must be a multiple of 24. Therefore, it is also a multiple of 12.
 Sufficient

Stat (2) 
Find the P.F. of 24.

>  24 = 2^3 × 3^1

Bump up the exponents again.

>  x^2  is a multiple of  2^4 × 3^2

>  x  is a multiple of  2^2 × 3^1 = 12

Thus,  x  must be a multiple of 12.
 Sufficient

(d) EACH statement ALONE is sufficient to answer the question asked.

deepalidmoon · Answer

Finally, Thank you!  shoyland

If x is a positive integer, is x a multiple of 12?

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