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If the number 13 completely divides x, and x = a2 * b, where a and b are distinct prime numbers, which of these numbers must be divisible by 169? 
  • a)
    a2  
  • b)
    b2
  • c)
    ab
  • d)
    ab2
  • e)
    a3
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
If the number 13 completely divides x, and x = a2 * b, where a and b a...
Step 1: Question statement and Inferences
 We are given that the number x is a multiple of 13.
 This means, x = 13 * k           (k is an integer)      …….. (1)
 Also,
 x = a2 * b,                    where a and b are prime numbers.             ………… (2)
 Now, since x has only two distinct prime factors a and b, and x is a multiple of 13, one of the numbers a and b is 13.
 So, either a = 13 or b =13. 
 We have to find the number from the given options that is definitely a multiple of 169.
 169 = 132
Step 2: Finding required values
 Now, if a = 13, we need the term a2 in the number to make it a multiple of 169. 
 And, if b = 13, we need the term b2 in the number to make it a multiple of 169.  
 We do not know which number out of a and b is equal to 13.
 So, we can only be sure that a given number is divisible by 132 if the powers of both a and b are 2 or higher in that number. In that case, whether a = 13 or b = 13, the number will be divisible by 132 for sure.
There is only one such number in the options: a2 b2.
Step 3: Calculating the final answer
 So, the number that is definitely a multiple of 169 is a2 b2.  
Answer: Option (D) 
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Most Upvoted Answer
If the number 13 completely divides x, and x = a2 * b, where a and b a...
To understand why the number a^2b^2 must be divisible by 169, let's break down the information given in the question step by step.

Given:
- The number 13 completely divides x.
- x = a^2 * b, where a and b are distinct prime numbers.

Step 1: Understanding Divisibility by 13
Since 13 completely divides x, it means that x is a multiple of 13. In other words, x is divisible by 13 without leaving any remainder.

Step 2: Analyzing the Prime Factorization of x
We are given that x = a^2 * b, where a and b are distinct prime numbers. This means that x can be expressed as the product of two prime numbers, a and b, raised to certain powers.

Step 3: Identifying Factors of x
Since x is divisible by 13, it means that 13 is a factor of x. Therefore, 13 must be one of the prime factors of x.

Step 4: Prime Factorization of 13
To find the prime factorization of 13, we need to check if it is divisible by any prime numbers less than 13. Since 13 is a prime number itself, it cannot be divided by any other prime numbers.

Therefore, the prime factorization of 13 is simply 13^1.

Step 5: Prime Factors of x
Since 13 is a factor of x, it means that 13 must be one of the prime factors of x. We can represent this as 13^1.

Step 6: Multiplying x by 13
To find the number that must be divisible by 169, we need to multiply x by 13. This can be represented as 13 * x.

Step 7: Simplifying the Expression
Since x = a^2 * b, we can substitute this expression into 13 * x to get 13 * (a^2 * b). By applying the distributive property, we can simplify this expression to (13 * a^2) * b.

Step 8: Recognizing the Divisible Number
From the simplified expression (13 * a^2) * b, we can see that (13 * a^2) is a factor of x. This means that (13 * a^2) must be divisible by 169, as 169 is the square of 13.

Therefore, the number a^2b^2 must be divisible by 169, making option D the correct answer.
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