Question

The product of two numbers is 1296. If one number is 16 times the other, find the two numbers?

Answer 1

Hint:
Here we need to find the value of two numbers whose product is equal to 1296. We will first assume the first term to be $x$ and then we will assume the second term to be $16x$. Then we will multiply both the assumed number and we will equate the given product. From there, we will get the value of $x$.

Complete step by step solution:
It is given that the product of two numbers is 1296 and the value of one of the numbers is 16 times the value of the other number.
Let the value of the first number be $x$ and the second number be $16x$.
We will find the product of these two numbers.
Product $=x\times 16x=16x^2$
We substitute the value of the product here.
Therefore,
$\Rightarrow 1296=16x^2$
On dividing 16 on both sides, we get
$\begin{array}{rl}& \Rightarrow {\displaystyle \frac{1296}{16}}={\displaystyle \frac{16x^2}{16}}\\ & \Rightarrow 81=x^2\end{array}$
Taking square roots on both sides, we get
$\begin{array}{rl}& \Rightarrow \sqrt{81}=\sqrt{x^2}\\ & \Rightarrow 9=x\end{array}$
Therefore, the first number is 9. Now, we will find the value of the second number which is 16 times the first number.
Therefore, 2nd number $=16\times 9=144$

Hence, the two numbers are 9 and 144.

Note:
Here the required numbers whose product is 1296 are 9 and 144. We can observe that both the numbers are a perfect square, where a perfect square is defined as a number which is made by squaring a whole number. Therefore, 9 is formed by squaring a whole number 3 and 144 is formed by squaring a whole number 12. Their product is 1296 which is also a perfect square. Always remember that a product of two or more perfect squares is also a perfect square.