Wednesday, 31 December 2025

Exercise (4.1).7

Show that

$$ 7^{40 353 606} \equiv 0 \pmod {40 353 607} $$

Is $40 353 607$ prime?


Trial and error leads us to

$7^9 \equiv 40353607 \equiv 0 \pmod {40353607}$

We also have $40353606 = 9 \times 4483734$. And so

$7^{40353606} \equiv (7^9)^{4483734} \equiv 0^{4483734} \equiv 0 \pmod {40353607}$


We have shown above that 40353607 has factors of 7, and so is not a prime.