Proof Of a problem of distinct prime numbers
We need to prove pq−1+qp−1≡ 1 mod (pq)
Solution:
Since p and q are different prime numbers, their greatest common divisor = 1
means (p, q) = 1
Here, we can Apply Fermat's little theorem, then
pq−1≡1 mod (q) and qp−1≡1 mod (p)
We know pq−1≡0 mod(p) and qp−1≡0 mod(q)
Now we can combine pq−1≡1 mod(q) and qp−1≡0 mod(q)
We get, pq−1+qp−1≡1 (mod q)
In the same way, we can also combine
qp−1=1 mod(p) and pq−1=0 (mod p)
qp−1+pq−1≡1 (mod q)
distinct prime numbers p and q both divides pq−1+qp−1−1
and also pq divides pq−1+qp−1−1
So, pq−1+qp−1−1≡0(mod pq)
pq−1+qp−1≡1 (mod pq)
Answer: pq−1+qp−1≡1 (mod pq)
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