The lcm and the hcf of two numbers are 1001 and 7 respectively. how many such pairs are possible?

Answer

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Hint: Here we will assume the two numbers to be any variables and then we will multiply them with the HCF. Then we will use the relation between the HCF and LCM of two numbers. From there, we will get all the possible pairs of numbers.

Complete step-by-step answer:

It is given that the LCM of the two numbers is equal to 400 and the HCF of the numbers is equal to 4.Since HCF of the numbers is 4. So, two numbers are multiples of 4.Let the numbers be \[4x\] and \[4y\].Now, we will use the relation between the HCF and LCM of two numbers.We know that the product of the given natural numbers is always equal to the product of HCF and LCM of the given natural numbers.Using this relation, we get\[4x \times 4y = 4 \times 400\]Multiplying the terms on both sides, we get\[ \Rightarrow 16xy = 1600\]Now, we will divide both sides by 16. Therefore, we get \[ \Rightarrow \dfrac{{16xy}}{{16}} = \dfrac{{1600}}{{16}}\]On further simplification, we get\[ \Rightarrow xy = 100\]The two numbers i.e. \[4x\] and \[4y\] are equal to 4 only when \[x\] and \[y\] are co prime numbers.So the possible pairs such that \[x\] and \[y\] are co prime numbers will be:-\[ \Rightarrow \left( {x,y} \right) = \left( {25,4} \right);\left( {100,1} \right)\]So the numbers will be \[\left( {4 \times 25,4 \times 4} \right) = \left( {100,16} \right)\] and \[\left( {4 \times 100,4 \times 1} \right) = \left( {400,4} \right)\].

Hence, there are only two possible pairs of the two numbers.

Note: To solve this question, we need to know the meaning of LCM and HCF and also the relation between them. LCM stands for the Lowest Common Factor and is defined as the smallest number that is a multiple of all these numbers. HCF stands for the Highest Common Factor and HCF of the given numbers is defined as the largest number that divides all these numbers completely. When the HCF of two integers or numbers is 1 then they are known as co prime numbers.

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