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

`(-4)/(c)* (-6)/(c)=6`

`(24)/(c^2)=6`

`(24)/(6)=c^2`

`4=c^2`

Taking square root on both sides, we get

`\pm √(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`.