Question

The product of two consecutive odd integers is 143. How do you find the integers?

Answer 1

11 and 13.

Explanation:

Call x a number
The number (2x + 1) should be an odd number no matter what x is.
The consecutive odd number to (2x + 1) will be: (2x + 3).
There for:
(2x + 1)(2x + 3) = 143
`4x^2 + 6x + 2x + 3 = 143`
`4x^2 + 8x - 140 = 0`
`x^2 + 2x - 35 = 0`
Solve this quadratic equation for x. Find 2 numbers knowing sum (-2) and product (-35). They are x = 5 and x = -7 (rejected as negative)
The odd number is N = (2x + 1) = 10 + 1 = 11
The consecutive odd number is: N + 2 = (2x + 3) = 10 + 3 = 13
Their product is (11 x 13) = 143. OK

Answer 2

Two solutions:

`11` and `13`

`-13` and `-11`

Explanation:

Let `n` be the even number between the two consecutive odd numbers `n-1` and `n+1`.

Then we have:

`143 = (n-1)(n+1) = n^2-1`

Add `1` to both ends to get:

`144 = n^2`

So `n = +-sqrt(144) = +-12`

If `n = 12` then the two odd numbers are `11` and `13`.

If `n = -12` then the two odd numbers are `-13` and `-11`.

Answer 3

Find `sqrt143`, then use trial and error

`11xx13 = 143" and "-11xx-13 = 143`

Explanation:

The consecutive odd numbers must lie on either side of the square root of 143

`sqrt 144 = 12` so `sqrt143 ~~ 11.9`

Look for odd numbers either side of `sqrt143`.

Try `11 and 13`

`11 xx 13 = 143" "larr` these are the factors we need.

Remember the factors could be negative as well..

`-11 xx -13 = 143`

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Background knowledge:
If you write the factors of a number in increasing order, note the following:

Factors are always in pairs.

• The difference between the factors is greatest for the outside factors and least for the innermost factors.

For example the factors of 48 are:

`1" "2" "3" "4" "6" "8" "12" "16" "24" "48`

48- 1 = 47 but 8-6 = 2

• The square root of a number is exactly in the middle of the factors

For example the factors of 36 are

`1" "2" "3" "4" "6" "9" "12" "18" "36`
`color(white)(.........................)uarr`
#color(white)(.......................)sqrt36

`6-6=0` this is the smallest possible difference.