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)