2021/045) Let $p>5$ be the prime number. Prove that the expression $p^4-10p^2+9$ is divisible by 1920?

 we need to show that $p^4 - 10p^2 +9)$ is divisible by 1920.

Let us now factor $p^4 - 10p^2 +9$

$p^4 - 10p^2 +9 = (p^2-1)(p^2-9)$

$= (p+1)(p-1)(p+3)(p-3)$

$= (p-3)(p-1)(p+1)(p+3)$

as p is greater than 5 and a prime so p is odd so let p = 2n + 1

so we get above expression

$=(2n-2)(2n)(2n+2)(2n+4) = 16(n-1)n(n+1)(n+2)$

$(n-1)n(n+1)(n+2)$ being product of 4 consecutive numbers is divisible by 24 so $16(n-1)n(n+1)(n+2)$ is divisible by 16 * 24 = 384

Further $(n-1)n(n+1)(n+2)= \frac{(n-1)n(n+1)(n+2)(n+3}{n+3}$ 

$(n-1)n(n+1)(n+2)(n+3)$ being product of 5 consecutive numbers is divisible by 5 but is p is prime so p is not

 divisible by 5 or 2n+1 is not divisible by 5 or 2n +6 is not divisible 5 or n +3 is not divisible by 5 so $(n-1)n(n+1)(n+2)$ is not divisible by 5

Hence $16(n-1)n(n+1)(n+2)$ is divisible by 5 and is it is divisible by 384 so divisible by 5 * 384 or 1920

so $p^4 - 10p^2 +9$ is divisible by 1920 hence proved.