# The product of two numbers is 1,360. The difference of the two numbers is 6. What are the two numbers?

Jun 5, 2018

40 and 34
OR
-34 and -40

#### Explanation:

Given that :
1) The product of two numbers is 1,360.

2) The difference of the two numbers is 6.

If the 2 numbers are $x$, and $y$

1) $\implies x \times y = 1360$

$\implies x = \frac{1360}{y}$

and 2) $\implies x - y = 6$

$\implies x = 6 + y$ ---------(i)

Substituting value of $x$ in 1),

$\implies \left(6 + y\right) y = 1360$

$\implies 6 y + {y}^{2} - 1360 = 0$

$\implies {y}^{2} + 6 y - 1360 = 0$

$\implies {y}^{2} + 40 y - 34 y - 1360 = 0$

$\implies y \left(y + 40\right) - 34 \left(y + 40\right) = 0$

$\implies \left(y - 34\right) \left(y + 40\right) = 0$

$\implies y = 34 \mathmr{and} y = - 40$

Taking $y = 34$, and finding value of $x$ from equation (2):

$x - y = 6$
$\implies x - 34 = 6$

$\implies x = 40$

So, $x = 40 \mathmr{and} y = 34$

or

If we take y= -40, then

2) $\implies x - \left(- 40\right) = 6$

$\implies x = 6 - 40 = - 34$

So, $x = - 40 , \mathmr{and} y = - 34$

Answer: The two numbers are : $40 \mathmr{and} 34$
OR
$- 34 \mathmr{and} - 40$

Jun 5, 2018

The numbers are $34 \mathmr{and} 40$

$34 \times 40 = 1360 \mathmr{and} 40 - 34 = 6$

#### Explanation:

Factors of a number are always in pairs. If you write them in ascending order there are several things we can observe.
For example: the factors of $36$.

$1 , \text{ "2," "3," "4," "6," "9," "12," "18," } 36$
$\textcolor{w h i t e}{\times \times \times \times x . \times x} \uparrow$
$\textcolor{w h i t e}{\times \times \times \times . \times x} \sqrt{36}$

The outer pair, $1 \mathmr{and} 36$ have a sum of $37$ and a difference of $35$, whereas the innermost pair, $6 \mathmr{and} 6$ have a sum of $12$ and a difference of $0$

The factor in the middle is $\sqrt{36}$. The further we are from the middle pair of factors, the greater is the sum and difference.

In this case, the factors of $1360$ only differ by $6$, which means that they are very close to its square root.

$\sqrt{1360} = 36.878 \ldots$

Explore numbers on either side of this. (Not more than $3 \mathmr{and} 4$ on either side.) You are also looking for factors which multiply to give a $0$ at the end.

$1360 \div 35 = 38.857$
$1360 \div 40 = 34 \text{ } \leftarrow$ here we have them!