Question

Two positive numbers have their HCF as \[12\] and their product as \[6336\]. Choose the correct option for the number of pairs possible for the given numbers.
A) \[2\]
B) \[3\]
C) \[4\]
D) \[5\]

Answer 1

Hint: Here we have to find the possibility of pairs. We will try to find the numbers. They give the HCF of two positive numbers and product of the two positive numbers. First we will consider the numbers in terms of HCF. From using the product of the integers, we can find the integers. Then we can find the required possibility.

Complete step-by-step answer:
It is given that the HCF of two positive integers is \[12\] and their product is \[6336\].
We have to find the possibility for the pairs of the given numbers.
Since, the HCF of the two positive numbers is \[12\], the numbers will be multiple of \[12\].
 Let us consider the two positive integers \[12x\] and \[12y\] where, \[x\] and \[y\] are co-prime numbers.
So, the product of these two numbers is \[144xy\].
It is given that the product of these two numbers is \[6336\].
According to the problem,
\[ \Rightarrow 144xy = 6336\]
Solving we get,
\[ \Rightarrow xy = \dfrac{{6336}}{{144}}\]
Simplifying we get,
\[ \Rightarrow xy = 44\]
The factors of \[44\] is \[1,{\text{ }}2,\;{\text{ }}4,\,{\text{ }}11,{\text{ }}22,{\text{ }}44\].
So, \[xy = 44\] can be written as the product of two factors in three ways.
These are: \[1 \times 44;{\text{ }}2 \times 22;\,{\text{ }}4 \times 11\]
Again, since, \[x\] and \[y\] are co-prime numbers, the possible value of \[x\] and \[y\] is \[x = 1\& y = 44\]or \[x = 4\& y = 11\]. We cannot take the option of \[x = 2\& y = 22\].
So, when \[x = 1\& y = 44\], the numbers are \[12\& 528\].
When \[x = 4\& y = 11\], the numbers are \[48\& 132\].
So, there is the possibility of two pairs.

$\therefore $ Hence, the correct option is A \[2\].

Note: A Co-prime number is a set of numbers or integers which have only 1 as their common factor i.e. their highest common factor (HCF) will be 1. Co-prime numbers are also known as relatively prime or mutually prime numbers.
The greatest number which divides each of the two or more numbers is called HCF or Highest Common Factor. It is also called the Greatest Common Measure (GCM) and Greatest Common Divisor (GCD).