# If the reciprocal of the product of the two consecutive integers is 1/30 how do you find the two integers?

Jul 5, 2015

Then the product must be the reciprocal of $1 / 30$ and that is $30$ (reciprocal goes both ways).

#### Explanation:

The reciprocal of the reciprocal is the original number.

So we need $n \cdot \left(n + 1\right) = 30$
You can try factoring $30$ in different ways, but it will be clear that only $5 \mathmr{and} 6$ satisfy the condition of being consecutive.

Jul 7, 2015

$5 , 6 \mathmr{and} - 6 , - 5$

#### Explanation:

*Integers:* The numbers which are not fractional(like $\frac{26}{9}$) are called integers.

Let A, B be two consecutive numbers.

suppose A=n say.

Then B must be "n+1" as it is next number to A.

Product of two consecutive numbers=AB.

The reciprocal of the product of the two consecutive integers=$\frac{1}{A B}$

But, the reciprocal of the product of the two consecutive integers is 1/30.

From equality rule,
Ultimately,

$\frac{1}{A B} = \frac{1}{30}$

Just substitute assumed A,B values.

$\frac{1}{n \left(n + 1\right)} = \frac{1}{30}$

$\implies \frac{1}{{n}^{2} + n} = \frac{1}{30}$

$\implies {n}^{2} + n - 30 = 0$

$\implies {n}^{2} + 6 n - 5 n - 30 = 0$

$\implies n \left(n + 6\right) - 5 \left(n + 6\right) = 0$

$\implies \left(n - 5\right) \left(n + 6\right) = 0$

$\implies n = 5 | n = - 6$

If $n = 5$;

$A = 5 ,$

$B = 6$

If $n = - 6 :$

$A = - 6$

$B = - 5$