Compute gcd$(24, 54 + 24^{7})$

Question

Compute $\gcd(24, 54 + 24^{7})$.

My attempt:

Euclidean algorithm:

$ a = bq + r$

$ 54 + 24^{7} = 24.24^{6} + 54$

In this case my $ \ r = 54 \gt 24 = b$. However, in Euclidean's algorithm $ \ 0 \le r < b$. What am I doing wrong?

Answer 1

$24^7+54=24.(24^6+2)+6$.

$24=6.4+0$.

So $\gcd(24^7+54,24)=\gcd(24,6)=\gcd(6,0)=6$.

Answer 2

Since $24^7$ divided by $24$, we see that $\gcd(24,54+24^7)=\gcd(24,54)=6.$