If both 11^2 and 3^3 are factors of the number a*4^3*6^2*13^11, then

Question

If both 11^2 and 3^3 are factors of the number  a*4^3*6^2*13^{11} , then what is the smallest possible value of a?

A. 33
B. 121
C. 363
D. 3267
E. None of the above

stonecold · Answer

Here C is correct as least value of a => must contain a 11^2 and a 3

udaypratapsingh99 · Answer

If a = 363, how come  3^3  is a factor of a*4^3*6^2*13^11 ??

luisdicampo · Answer

To find the smallest possible value of  a , we need to analyze the prime factorization of the given number and compare it with the required factors.

Step 1: Prime Factorization of the Given Number 
The number is given as:  N = a \cdot 4^3 \cdot 6^2 \cdot 13^{11}

Let's break the constant terms into their prime bases:
 4^3 = (2^2)^3 = 2^6 
 6^2 = (2 \cdot 3)^2 = 2^2 \cdot 3^2 
 13^{11}  is already in prime base.

Now, combine these parts:
 N = a \cdot (2^6) \cdot (2^2 \cdot 3^2) \cdot (13^{11}) 
 N = a \cdot 2^8 \cdot 3^2 \cdot 13^{11}

Step 2: Identify the Missing Factors 
The problem states that both  11^2  and  3^3  must be factors of  N .

Let's check what we already have versus what we need:

1.  Requirement:  11^2  
 Does our current expression ( 2^8 \cdot 3^2 \cdot 13^{11} ) contain any factors of 11?
  No. 
 Therefore,  a  must provide the entire  11^2 .

2.  Requirement:  3^3  
 Does our current expression contain factors of 3?
  Yes , it has  3^2 .
 However, we need  3^3 .
 We are missing exactly one factor of 3 ( 3^3 - 3^2 \rightarrow  need  3^1 ).
 Therefore,  a  must provide at least  3^1 .

Step 3: Calculate the Smallest 'a' 
To find the smallest possible value,  a  should consist  only  of the missing factors.

a = 11^2 \cdot 3^1 
 a = 121 \cdot 3 
 a = 363

This matches Option (C).

Answer: C

If both 11^2 and 3^3 are factors of the number a4^36^2*13^11, then

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