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

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

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Answer 1 · 2016-12-25T13:32:44

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
$4 x^{2} + 6 x + 2 x + 3 = 143$ $4x^2 + 6x + 2x + 3 = 143$
$4 x^{2} + 8 x - 140 = 0$ $4x^2 + 8x - 140 = 0$
$x^{2} + 2 x - 35 = 0$ $x^2 + 2x - 35 = 0$
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 · 2016-12-25T14:52:31

Two solutions:

$11$ $11$ and $13$ $13$

$- 13$ $-13$ and $- 11$ $-11$

Explanation:

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

Then we have:

$143 = (n - 1) (n + 1) = n^{2} - 1$ $143 = (n-1)(n+1) = n^2-1$

Add $1$ $1$ to both ends to get:

$144 = n^{2}$ $144 = n^2$

So $n = \pm \sqrt{144} = \pm 12$ $n = +-sqrt(144) = +-12$

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

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

Answer 3 · 2016-12-25T16:24:11

Find $\sqrt{143}$ $sqrt143$ , then use trial and error

$11 \times 13 = 143 and - 11 \times - 13 = 143$ $11xx13 = 143" and "-11xx-13 = 143$

Explanation:

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

$\sqrt{144} = 12$ $sqrt 144 = 12$ so $\sqrt{143} \approx 11.9$ $sqrt143 ~~ 11.9$

Look for odd numbers either side of $\sqrt{143}$ $sqrt143$ .

Try $11 and 13$ $11 and 13$

$11 \times 13 = 143 \leftarrow$ $11 xx 13 = 143" "larr$ these are the factors we need.

Remember the factors could be negative as well..

$- 11 \times - 13 = 143$ $-11 xx -13 = 143$

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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:

$12346812162448$ $1" "2" "3" "4" "6" "8" "12" "16" "24" "48$

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

$123469121836$ $1" "2" "3" "4" "6" "9" "12" "18" "36$
$color(white)(.........................)uarr$
#color(white)(.......................)sqrt36

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

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

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