Question

The product of two successive natural numbers is $1980$ . Which is the smaller number?

Answer 1

Hint: For solving these problems, we need to have a complete understanding of what are natural numbers and how to convert a statement into a mathematical equation. The above problem can be converted into a quadratic equation. Thus, by solving the equation and identifying the natural number, we can find the answer.

Complete step-by-step solution:
Natural numbers are all numbers \[1,2,3,4...\] . They are the numbers one usually counts and they will continue on into infinity. Two consecutive or successive natural numbers are those which are next to each other. i.e., \[2,3\] or \[6,7\] or \[9,10\] and so on. The difference between them is \[21,76,109\] , which are all the same namely $1$ . So, if we consider a natural number to be x then its successive natural number will be \[x+1\] , where x is the smaller number and \[x+1\] is the larger number.
According to the given problem, the product of two successive natural numbers is given as $1980$ . If we write this statement mathematically in the form of equation, we get that
\[x\left( x+1 \right)=1980\]
Now by solving, we get the quadratic equation as ${{x}^{2}}+x-1980=0$ . We need to find the roots of this equation. By employing Sridhar Acharya’s formula, we get that,
$\begin{align}
  & x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \\
 & \Rightarrow x=\dfrac{-1\pm \sqrt{{{1}^{2}}-4\left( -1980 \right)}}{2} \\
 & \Rightarrow x=\dfrac{-1\pm 89}{2}=44,-45 \\
\end{align}$
Now according to the given problem, x is a natural number hence we have to eliminate \[-45\] due to the fact that it is not a natural number. Hence the natural number is $44$ and \[x+1=45\] is its successive natural number whose product gives $1980$ . Therefore, the smaller natural number is $44$.

Note: These types of problems are pretty easy to solve but a slight misjudgement in the calculation can lead to a totally different answer. This can also be solved by breaking the number into two successive natural numbers using factorization and thereby we can easily identify the smaller number.