Question

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

Answer 1

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) `=> x xx y =1360`

`=> x = 1360/y`

and 2) `=> x-y =6`

`=> x= 6+y` ---------(i)

Substituting value of `x` in 1),

`=> (6+ y) y = 1360`

`=> 6y + y^2 -1360 =0`

`=> y^2 +6y -1360 =0`

`=> y^2 +40y -34y -1360 =0`

`=> y(y +40) - 34(y+40) = 0`

`=> (y-34)(y+40) =0`

`=> y=34 or y= -40`

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

`x-y =6`
`=> x - 34 = 6`

`=> x= 40`

So, `x= 40 and y =34`

or

If we take y= -40, then

2) `=> x- (-40) =6`

`=> x= 6 - 40 = -34`

So, `x= -40, and y= -34`

Answer: The two numbers are : `40 and 34`
OR
`-34 and -40`

Answer 2

The numbers are `34 and 40`

`34 xx 40 = 1360 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," "2," "3," "4," "6," "9," "12," "18," "36`
`color(white)(xxxxxxxxx.xxx)uarr`
`color(white)(xxxxxxxx.xxx)sqrt36`

The outer pair, `1 and 36` have a sum of `37` and a difference of `35`, whereas the innermost pair, `6 and 6` have a sum of `12` and a difference of `0`

The factor in the middle is `sqrt36`. 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.

`sqrt1360 = 36.878...`

Explore numbers on either side of this. (Not more than `3 or 4` on either side.) You are also looking for factors which multiply to give a `0` at the end.

`1360 div35 = 38.857`
`1360 div 40 = 34" "larr` here we have them!