Solutions To Some Sastry Problems On Smarandache Number Related Triangles

In his recent paper[l], Sastry defines two triangles T(a,b,c) and T(a',b',c') to be Smarandache related ifS(a) = Sea'), S(b) = S(b') and S(c) = S(c'). The function S is known as the Smarandache function and is defmed in the following way. For n any integer greater than zero, the value of the Smarandache function Sen) is the smallest integer m such that n divides m!. He closes the paper by asking the following questions: A) Are there two distinct dissimilar Pythagorean triangles that are Smarandache related? A triangle T(x,y,z) is Pythagorean ifx*x + y*y = z*z. B)Are there two distinct and dissimilar 60(120) degrees triangles that are Smarandache related? A 60(120) degrees triangle is one containing an angle of 60(120) degrees. C) Given a triangle T(a,b,c), is it possible to give either an exact formula or an upper bound for the total number of triangles (without actually determining them), which are Smarandache related to T? D) Consider other ways of relating two triangles in the Smarandache number sense. For example, are there two triplets of natural numbers (a,b,c) and (a',b',c') such that a + b + C = a' + b' + c' = 180 and Sea) = Sea'), S(b) = S(b') and S(c) = S(c')? If this were true, then the angles, in degrees, of the triangles would be Smarandache related. In this paper, we will consider and answer questions (A), (B) and (D). Furthermore, we will also explore these questions using the Pseudo Smarandache function Zen). Given any integer n >0, the value of the Pseudo Smarandache function is the smallest integer m such that n evenly divides m L: k. k=l A) The following theorem is easy to prove.