Question

How many ordered pairs m n Satisfy the equation mn+2m-2n=2020 if m,n E N

Answer 1

To find:

How many ordered pairs $(m,n)$ satisfy the equation $mn+2m-2n=2020$ if $m,n\in\mathbb{N}$?

Step-by-step explanation:

Given, $mn+2m-2n=2020$

$\Rightarrow mn+2m-2n-4=2016$

$\Rightarrow m(n+2)-2(n+2)=2016$

$\Rightarrow (n+2)(m-2)=2016$ ... ... (1)

Let's prime factorize $2016$.

Refer to the attachment.

Thus, $2016=2\times 2\times 2\times 2\times 2\times 3\times 3\times 7$

Let us use the above factors of $2016$ to find $m,n$ from (1) no. relation.

• When $m=4,\:n=504$

• When $m=8,\:n=252$

• When $m=16,\:n=126$

• When $m=32,\:n=63$

• When $m=96,\:n=21$

• When $m=288,\:n=7$

Answer:

Thus we have obtained six pairs of $(m,n)$ that satisfy the given equation.

Answer 2

Answer:

Im just adding the answer given above:

.

actual pairs will be (6,502) , (10,250) , (18,124) , (34,61) , (98,19) , (290,5)

.

also you can take prime factors in different order to get more such pairs.

one such example would be....

first factor = 2×7 = 14

second factor = 2×2×2×2×3×3 = 144

hence (16,142).