Now, we are to find the smallest integer value of $ t $ such that $ S(t) $ is an integer. Since $ S(t) = t + 3 $ for all $ t
["Finding the Smallest Integer Value of ( t ) Such That ( S(t) = t + 3 ) is an Integer", "In many mathematical problems, identifying specific values for variables is key to solving functional expressions. In this case, we are given a simple function:", "[\nS(t) = t + 3\n]", "where ( t ) is any integer. The question asks: What is the smallest integer value of ( t ) such that ( S(t) ) is an integer?", "### Understanding the Function", "The function ( S(t) = t + 3 ) takes an integer input ( t ) and adds 3, producing an output that is always an integer whenever ( t ) is an integer. Since 3 is itself an integer, adding ( t ) (an integer) to it guarantees that the result will always be an integer.", "### Solving for the Smallest Integer ( t )", "We seek the smallest integer ( t ) such that ( S(t) ) is an integer. However, because ( S(t) ) is defined as ( t + 3 ), and both ( t ) and 3 are always integers for all integer ( t ), the expression ( S(t) ) is guaranteed to be an integer for any integer ( t ).", "Thus, mathematically, the property holds for all integers, including the smallest possible integers. Since there is no restriction (like ( t ) being positive or non-negative), the smallest integer value of ( t ) is:", "[\nt = -\infty \quad \ ext{(infinite descending)}\n]", "But in practical terms, when asking for the "smallest integer," we typically consider the lowest finite integer meeting the condition. However, because the function is defined and outputs integers for all integers, there is no minimum finite integer—the set of integers extends infinitely downward.", "Yet, in contexts such as algorithmic inputs, practical constraints, or problem design, if we interpret “smallest” as the least integer value that satisfies the mathematical condition without restriction, we conclude:", "[\n\ ext{There is no smallest integer } t \ ext{ satisfying } S(t) \in \mathbb{Z}, \ ext{ because } S(t) \ ext{ is always an integer for all integers } t.\n]", "But if the intent is to pick the smallest integer value (allowing negative integers), the answer embraces the infinite negative range. However, in typical mathematical wording and olympiad-style problems, when such flexibility exists and the function outputs integers for all integers, the phrase “smallest” may imply the condition holds universally—so we note:", "The smallest integer value does not exist in a finite sense—all integers satisfy the condition, so every integer is valid, including infinitely small ones.", "### Practical Clarification", "In real-world or competition contexts, if ( t \in \mathbb{Z} ) and ( S(t) = t + 3 ) is required to be an integer, the condition holds automatically. Therefore, every integer is a valid solution, and no smallest value exists in the conventional number line sense.", "However, if the problem intends to restrict ( t \geq 0 ) or another constraint, please specify. Otherwise, the mathematical truth is:", "[\n\boxed{t \ ext{ can be any integer; thus, there is no smallest integer, but } S(t) \ ext{ is always an integer for all integer } t.}\n]", "For educational clarity: since ( S(t) = t + 3 ) preserves integrality for all integer inputs, any integer ( t ) satisfies the condition. There is no minimal such ( t ); the set ( \mathbb{Z} ) has no lower bound.", "---", "Forstrong integer functionality:\nBecause addition of integers preserves integrality, and ( t \in \mathbb{Z} \implies t + 3 \in \mathbb{Z} ), the condition holds universally. Hence, the solution set is all integers—so no singular “smallest” value exists."]
