Question

Product of two natural numbers p and q is 590 and their hcf is 59. how many set of values of p and q is possible

Answer 1

$\text{Let the product of two natural numbers p and q is 590, and their HCF is 59}\\ \\ \text{we know that the product of LCM and HCF of any two numbers is equal}\\ \text{to the product of the numbers. that is}\\ \\ \text{HCF}\times \text{ LCM}=p\times q\\ \\ \Rightarrow 59 \times \text{LCM}=590\\ \\ \Rightarrow \text{LCM}=\frac{590}{59}\\ \\ \Rightarrow \text{LCM}=10\\ \\ \text{for any two natural numbers, their Least Common Multiple (LCM) is always}$

$\text{greater than their HCF.}\\ \\ \text{but here we can see that }LCM$

Hence there is no such natural numbers exist.