Find the smallest positive integer k satisfying the stated condition or explain no such k exists: 2^k -= 1 mod 14.

Question

Find the smallest positive integer k satisfying the stated condition or explain no such k exists:

2^{k} \equiv 1 mod 14

.

Answer 1

Find $2^{k} (mod 14)$ for some values of k.
$2^{1} \equiv 2 (mod 14)$
$2^{2} \equiv 4 (mod 14)$
$2^{3} \equiv 8 (mod 14)$
$2^{4} \equiv 2 (mod 14)$
$2^{5} \equiv 4 (mod 14)$
......
So, $2^{k} (mod 14)$ gives the value 2,4 or 8 for any positive integer k.
Hence, there exists no positive integer k such that $2^{k} \equiv 1 (mod 14)$ .

Find the smallest positive integer k satisfying the stated condition or explain no such k exists: 2^k -= 1 mod 14.

Want to know more about Congruence?

Expert Answer

Not exactly what you’re looking for?

Relevant Questions