Question

How do you find two consecutive positive odd integer numbers whose product is 143?

Answer 1

Hint: To solve this problem, consider two unknowns with a difference of 2 that is x and x+2. After that, form an equation by multiplying the two unknown variables and equating it with 143. Now, solve the formed equation using the general rules to solve a quadratic equation to find the two possibilities -13 and 11. Since, the questions need positive integers we will ignore -13 and consider 11 as our first integer. And if 11 is our x then our second digit will be x+2 that is 13.

Complete step-by-step answer:
The aim of our question is to find two consecutive positive odd integers whose product is 143
So, just because they have mentioned there are two consecutive odd numbers, therefore the difference of them will be 2 always. So, that’s why let’s suppose the numbers are $ x $ and $ x + 2 $ .
Now, it is stated that
 $ x \times (x + 2) = 143 $
All we have to do now is to simplify this equation to get the answer. So,
 So, we have ourselves a quadratic equation and all we have to do is to solve this to get the value of x which satisfies the equation
Following the general rules for solving a quadratic equation, we will first try to find two numbers whose sum equals the coefficient of x that is 2 and product equal to -143
By trial and error method we can say that those numbers are 11 and 13
Therefore, after modifying the equation it will become:-
 $
{x^2} + 2x - 143 = 0 \\
\Rightarrow {x^2} + 13x - 11x - 143 = 0 \;
 $
Taking x common from first two terms and -11 common from the last two terms
 $ {x^2} + 13x - 11x - 143 = 0 \\
\Rightarrow x(x + 13) - 11(x + 13) = 0 \\
\Rightarrow (x - 11)(x + 13) = 0 \;
 $
Now, we know whenever the product of any two terms is 0, either or both of the numbers must be 0
Therefore,
Either $ (x + 13) = 0 $ or $ (x - 11) = 0 $
If
 $
(x + 13) = 0 \\
\Rightarrow x = - 13 \;
 $
If
 $
x - 11 = 0 \\
\Rightarrow x = 11 \;
 $
But we know the two numbers are positive integers, hence we will ignore the -13 value.
So, $ x = 11 $ will be our first integer
And $ x + 2 = 13 $ will be our second integer
So, the correct answer is “ 11 and 13 ”.

Note: The multiplication done while forming the equation must be done carefully. Also, you are supposed to take simple unknown variables like x and x+2 to calculate the answers. Taking values like x and y or x+2 and x+4 will add on to the calculation and at times make the question unsolvable.