If the polynomial $R (x)$ is the remainder upon dividing $x^{2020}$ by $x^{2} - 7 x + 12$ . If $R (1)$ can be expressed as $(p^{m} - q^{n})$ , where $m, n \in N, and p & q$ are prime numbers, then the value of $n + m - p q$ is

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Solution

Let $R (x) = a x + b$
Then $x^{2020} = p (x^{2} - 7 x + 12) + R (x)$
$\Rightarrow x^{2020} = (x - 3) (x - 4) f (x) + a x + b . . . (1) where f (x) is a polynomial function$

Putting $x = 3$ in equation $1$
we get $3^{2020} = 3 a + b . . . (2)$

Putting $x = 4$ in equation $1$
we get $4^{2020} = 4 a + b . . . (3)$

From $2$ and $3$
we get $a = 4^{2020} - 3^{2020}$
$b = 4 \cdot 3^{2020} - 3 \cdot 4^{2020}$

$R (1) = a + b$
$\Rightarrow 4^{2020} - 3^{2020} + 4 \cdot 3^{2020} - 3 \cdot 4^{2020} = 3^{2021} - 2^{4041}$

equating it with the expression we get
$n + m - p q = 4041 + 2020 - 6 = 6056$

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If the polynomial R(x) is the remainder upon dividing x2020 by x2−7x+12. If R(1) can be expressed as (pm−qn), where m,n∈N, and p & q are prime numbers, then the value of n+m−pq is

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