This paper will examine Local Singular Arithmetical Congruence Monoids (ACM's) and determine specific properties that lead to the ACM being fully elastic. We develop a constructive process to investigate this. We break ACM's into two distinct classes. First, we will restrict our view to a submonoid of the given ACM which is generated by two distinct prime numbers. For the first class, the prime numbers are carefully chosen so that the submonoid is fully elastic on some open interval. In less complicated cases, the interval spanned by our submonoid covers the original monoid's interval yielding full elasticity for the monoid. In more complicated cases, we define multiple submonoids that span portions of the entire moniod's interval, then we take a union of the submonoids to create a covering of the monoid's interval again yielding full elasticity. For the second class, we apply the same approach and discuss where this approach is successful and where it fails. We then apply numerical results to specific cases and make conjectures based on these results.
