Question

James is solving a number puzzle that involves three integers, a, b, and c, where c is a positive integer. The product of a and b is 6. The product of a and c is -4. The product of b and c is -6.

Answer 1

Given:

Three integers, a, b, and c, where c is a positive integer.

The product of a and b is 6.

The product of a and c is -4.

The product of b and c is -6.

To find:

The values of a,b and c.

Solution:

According to the given information:

$ab=6$        ...(i)

$ac=-4$       ...(ii)

$bc=-6$      ...(iii)

From (ii), we get

$a=-\dfrsc{4}{c}$          ...(iv)

From (iii), we get

$b=-\dfrsc{6}{c}$         ...(v)

Putting $a=-\dfrsc{4}{c}$  and $b=-\dfrsc{6}{c}$ in (i), we get

$\dfrac{-4}{c}\times \dfrac{-6}{c}=6$

$\dfrac{24}{c^2}=6$

$\dfrac{24}{6}=c^2$

$4=c^2$

Taking square root on both sides, we get

$\pm \sqrt{4}=c$

$\pm 2=c$

It is given that c is a positive integer. So, it cannot be negative and the only value of c is $c=2$.

Putting $c=2$ in (iv), we get

$a=-\dfrsc{4}{2}$

$a=-2$

Putting $c=2$ in (v), we get

$b=-\dfrsc{6}{2}$

$b=-3$

Therefore, the values of a,b,c are $a=-2,b=-3,c=2$.