What is the smallest integer x such that 6^x is a multiple of 144 ?

Question

A. 2
B. 3
C. 4
D. 5
E. 6

$!vakumar.m · Answer

What is the smallest integer x such that 6^x (2*3)^x is a multiple of 144 ?

factors of 144 are - 2^4 * 3^2
so smallest integer x for 6^x to be multiple of 144 is x =4. i.e., 6^4 = 1296.
Answer C

Abhishek009 · Answer

6^x = 12^2

6^x = 2^2*6^2

Thus, x = 2, hence the multiple of 6^x as multiple of 144 is 4, Answer must be (C)

rito54 · Answer

Hello! Just double checking here it because we have the 2^4 which makes it obligated for x to be equal to 4? Thanks

Bunuel · Answer

_______________________
Correct.

luisdicampo · Answer

Deconstructing the Question  
We need to find the smallest integer  x  such that  6^x  is a multiple of 144.
Condition:  \frac{6^x}{144} = 	ext{Integer} .

Step 1: Prime Factorization  
Break down the numbers into their prime bases.
* Base:  6 = 2 \cdot 3 \implies 6^x = 2^x \cdot 3^x 
* Divisor:  144 = 12^2 = (2^2 \cdot 3)^2 = 2^4 \cdot 3^2

Step 2: Compare Exponents  
For  6^x  to be divisible by 144, the exponents of the prime factors in the numerator must be greater than or equal to the exponents in the denominator.

\frac{2^x \cdot 3^x}{2^4 \cdot 3^2}

We derive two inequalities:
1. For base 2:  x \ge 4 
2. For base 3:  x \ge 2

Step 3: Determine the Solution  
To satisfy both conditions,  x  must be at least the larger of the two required values.
The "bottleneck" is the power of 2, which requires  x  to be at least 4.

Therefore, the smallest integer  x  is 4.

Answer:  C

What is the smallest integer x such that 6^x is a multiple of 144 ?

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What is the smallest integer x such that 6^x is a multiple of 144 ?

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