A number is a pseudoprime

Question

Can Someone please help me prove that the number 1729 is a Pseudoprime?

So a pseudoprime is a composite $n$ such that $n |(2^n − 2)$. And every prime number also has this property.

Answer 1

$2^{1729}-2=2(2^{1728}-1).$ And $1729=7\cdot 13 \cdot 19.$ Check that each of $6,12,18$ (each one less than the primes $7,13,19$) divide the exponent $1728.$ So (using Fermat's "Little Theorem" here) $2^{1728}-1$ is divisible by the product $7 \cdot 13 \cdot 19 = 1729.$ Hence if we double this we get $1729$ being a divisor of $2^{1729}-2.$