IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 18, Issue 4 Ser. IV (Jul. – Aug. 2022), PP 01-08 www.iosrjournals.org DOI: 10.9790/5728-1804040108 www.iosrjournals.org 1 | Page The Form of The Friendly Number of 10 Sourav Mandal souravmandal1729@gmail.com Abstract Any positive integer  other than 10 with abundancy index 9 5 ⁄ must be a square with at least 6 distinct prime factors, the smallest being 5, and my new argument about the form of the friendly number of 10 is 25(5 2+1 −1)(8+1−2) 4 , if exist. Further at least one of the prime factors must be congruent to 1 modulo 3 and appear with an exponent congruent to 2 modulo 6 in the prime factorization of . --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 12-08-2022 Date of Acceptance: 27-08-2022 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction For a positive integer , the sum of the positive divisors of  is denoted by();the ratio ()  is known as the abundancy index of n or sometimes the ratio denoted as  () or ().A pair (a,b) is called a friendly pair if  () =  () in this case, it is also common to say that b is a friend of  or simply that  and  are friend. Perfect numbers have abundancy index 2, and thus all friendly numbers with abundancy index less than 2 are often called deficient, while numbers whose abundancy index are greater than 2 are called abundant. The original problem was to show that the density of friendly integers ℕ,is unity and the density of solitary numbers (numbers with no friends) is zero. We used two main approaches: one was an analysis of the ()  function. While the other used number theoretic arguments to find a representation for a friend of 10.In[1], it was shown that 10 is the smallest number where it is unknown whether there are any friends of it. We will assume a basic understanding of the function (). This can be found in [2]. For more on number theoretic techniques, see [3].And the question “Is 10 a solitary number” is still unanswered. If 10 does have a friend, the following may be of use in finding it. 1. Elementary Properties of Abundancy Index: Let  and  be positive integers. In what follows, all primes are positive. 1. () ≥ 1with equality only if =1 2. If  divides  then () ≤ () with equality only if = 3. If  1 , 2 ,……,  are distinct primes and  1 , 2 ,……,  are positive integers then  (∏      =1 ) = ∏(∑   −   =0 )  =1 =∏     +1 −1     (  − 1)  =1