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Question: The product of two successive natural numbers is \(1980\) . Which is the smaller number?...

The product of two successive natural numbers is 19801980 . Which is the smaller number?

Explanation

Solution

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...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., 2,32,3 or 6,76,7 or 9,109,10 and so on. The difference between them is 21,76,10921,76,109 , which are all the same namely 11 . So, if we consider a natural number to be x then its successive natural number will be x+1x+1 , where x is the smaller number and x+1x+1 is the larger number.
According to the given problem, the product of two successive natural numbers is given as 19801980 . If we write this statement mathematically in the form of equation, we get that
x(x+1)=1980x\left( x+1 \right)=1980
Now by solving, we get the quadratic equation as x2+x−1980=0{{x}^{2}}+x-1980=0 . We need to find the roots of this equation. By employing Sridhar Acharya’s formula, we get that,
x=−b±b2−4ac2a ⇒x=−1±12−4(−1980)2 ⇒x=−1±892=44,−45 \begin{aligned} & 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{aligned}
Now according to the given problem, x is a natural number hence we have to eliminate −45-45 due to the fact that it is not a natural number. Hence the natural number is 4444 and x+1=45x+1=45 is its successive natural number whose product gives 19801980 . Therefore, the smaller natural number is 4444.

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.