2022/009) Find the gcd(19! + 19, 20! + 19)

 We have $GCD(19! + 19, 20!+19)$ 

$= GCD( 19!+ 19 , 20(19! + 19) - (20!+ 19))$ using gcd(a,b) = gcd(a, na -b))

$= GCD(19! + 19, 20! + 20 * 19 - 20! -19)$

$= GCD(19! + 19, 20 * 19 - 19)$

$= GCD(!9! + 19, 361)$

Now $361 = 19^2$

$19! + 19 $is 1$9(18! + 1)$

As 19 is a prime as per wilson's theorem 18! + 1 is divisible by 19

So $19! + 19$ is divisible by $19^2 = 361$

so GCD = 361