If a sum of 2 natural numbers is 8 and their product is 15. Find the numbers.
ANSWER
Verified
Hint: Assume two variables for the two numbers and write the equation representing their sum and product. Solve the two equations in two variables to get the value of the numbers.
Complete step-by-step answer: Natural numbers are positive numbers starting from 1 and going endlessly. It is represented by \[\mathbb{N}\]. It forms the counting numbers. It is given that the two numbers are natural numbers, hence, they are positive numbers and greater than zero. Let the two natural numbers be x and y respectively. The sum of these two natural numbers is 8. Hence, we have: \[x + y = 8\] Let us write y in terms of x to get as follows: $\Rightarrow$ \[y = 8 - x............(1)\] Then, it is also given that the product of these two natural numbers is 15. Then, we have: \[xy = 15...........(2)\] Substituting equation (1) in equation (2), we get as follows: $\Rightarrow$ \[x(8 - x) = 15\] Simplifying the above equation, we have: $\Rightarrow$ \[8x - {x^2} = 15\] Taking all terms to one side of the equation, we have: $\Rightarrow$ \[{x^2} - 8x + 15 = 0\] We can solve this quadratic equation by splitting the middle term. –8x can be written as the sum of – 3x and – 5x, then, we have: $\Rightarrow$ \[{x^2} - 3x - 5x + 15 = 0\] Taking common terms and simplifying, we get: $\Rightarrow$ \[x(x - 3) - 5(x - 3) = 0\] Taking (x – 3) as the common term, we get: $\Rightarrow$ \[(x - 5)(x - 3) = 0\] The solutions are as follows: $\Rightarrow$ \[x = 5; x = 3\] For x = 5, using this in equation (1), we get y = 3. For x = 3, using this in equation (1), we get y = 5. Hence, the two natural numbers are 3 and 5.
Note: You can also solve for y first and then substitute to find the value of x, the result is invariant of the order in which we find the variables.