So $k = 7m + 2$. Then $x = 60k = 60(7m + 2) = 420m + 120$. We seek the smallest three-digit such $x$:
["# Solving for the Smallest Three-Digit Value of ( x = 60k ), Where ( k = 7m + 2 )", "Understanding how to systematically find specific values generated by algebraic expressions is a valuable skill in mathematics and algorithmic problem solving. This article explores the expression ( x = 60k ) with a twist—where ( k = 7m + 2 )—and identifies the smallest three-digit value of ( x ).", "---", "## The Formula Breakdown", "We start with:", "[\nk = 7m + 2\n]", "Since ( x = 60k ), substitute ( k ):", "[\nx = 60(7m + 2)\n]", "Distribute the 60:", "[\nx = 420m + 120\n]", "Now, our goal is to find the smallest three-digit integer value of ( x ). That is, the smallest ( x ) such that:", "[\n100 \leq x < 1000\n]", "---", "## Finding the Minimum ( m ) to Satisfy the Inequality", "We solve:", "[\n420m + 120 \geq 100\n]", "Subtract 120 from both sides:", "[\n420m \geq -20\n]", "Divide by 420:", "[\nm \geq -\frac{20}{420} = -\frac{1}{21}\n]", "Since ( m ) must be an integer (as it typically represents a variable in number-theoretic contexts), the smallest possible integer ( m ) satisfying this inequality is:", "[\nm = 0\n]", "---", "## Calculating ( x ) at ( m = 0 )", "Plug ( m = 0 ) into the expression for ( x ):", "[\nx = 420(0) + 120 = 120\n]", "So, ( x = 120 ) is a candidate—but is it the smallest three-digit value?", "Check:\n- ( 120 ) is a three-digit number — yes.\n- Is any smaller value possible with integer ( m )? Try ( m = -1 ):", "[\nx = 420(-1) + 120 = -300 \quad \ ext{(not valid, not ≥ 100)}\n]", "Thus, the smallest valid ( m ) gives the smallest valid ( x = 120 ), which is a three-digit number.", "---", "## Verifying Minimality", "We confirm:\n- ( x = 120 )\n- ( x = 420m + 120 ) increases as ( m ) increases\n- The next value for ( m = 1 ) is ( 420(1) + 120 = 540 )\n- Smaller three-digit values (e.g., 100 to 119) never satisfy ( x = 420m + 120 ) for any integer ( m )", "Hence, 120 is indeed the smallest three-digit value of ( x ).", "---", "## Interpretation and Applications", "Expressions like ( x = 60k ) with modular constraints come up in number theory, algorithm design, and cryptography. The structure ( x = 60(7m + 2) ) ensures ( x \equiv 120 \pmod{420} ), so all solutions are spaced 420 apart starting from 120. Recognizing this pattern allows efficient computation and constraints modeling.", "---", "## Conclusion", "To find the smallest three-digit value of ( x ) satisfying ( x = 60k ) with ( k = 7m + 2 ):", "- Substitute and simplify to ( x = 420m + 120 )\n- Solve for smallest integer ( m ) such that ( x \geq 100 )\n- Found ( m = 0 \Rightarrow x = 120 )\n- Confirmed no smaller valid ( x ) exists", "Final Answer: The smallest three-digit value of ( x ) is 120.", "---", "## Key Takeaways", "- Substitution and algebraic simplification streamline complex expressions.\n- Modular or linear constraints often yield arithmetic progressions.\n- Testing boundary values around computed minima ensures accuracy.\n- Known algebraic mappings reduce trial-and-error effort in problem-solving.", "---", "Keywords: ( k = 7m + 2 ), ( x = 60k ), smallest three-digit ( x ), algebraic expression, modular arithmetic, number theory, linear Diophantine values, optimization in integers."]
