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.

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:

       ...(i)

      ...(ii)

     ...(iii)

From (ii), we get

         ...(iv)

From (iii), we get

        ...(v)

Putting  and in (i), we get

Taking square root on both sides, we get

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

Putting in (iv), we get

Putting in (v), we get

Therefore, the values of a,b,c are .

check the picture below.

well, the angle at the vertex d is clearly not a right-angle, let's check for the one at the vertex f which does look like,   however if it's indeed, that means ef is perpendicular to df, in which case their slopes will be negative reciprocal of each other.

so the negative reciprocal of the slope of ef is 1, however the slope of df is not 1, is 1/2, so nope, they're not perpendicular lines.