On The Friendly Number Of 10

To answer the question "Is $10$ a solitary number?", we have established a necessary and sufficient condition on primes p and q so that p divides σ(q²ᵃ)(a⁺). Also We have found the form of q whenever p divides σ(q²ᵃ) for $q>p$ . Using these two, we have proved that if n is a friend of $10$ then it is a odd square with at least seven distinct prime factors, $5$ being the least one. Moreover there exists a prime factor p of n such that 2a+1≡ 0 f and 5ᶠ≡ 1 p where f is the smallest odd positive integer greater than $1$ and less than or equal to min\ 2a+1,p-1\, provided 5²ᵃ n, a∈ Z⁺. Further, there exist prime factors p and q not necessarily distinct of n such that p≡1 10 and q≡ 1 6. Besides, we have proved that if a Fermat prime Fₖ divides n then n must have a prime factor congruent to $1$ modulo 2Fₖ.