The sum of three consecutive integers is 51. What is the smallest of these integers?

The smallest integer equals 16 when the three consecutive integers are algebraically represented as n, n+1, and n+2, their sum is expressed as 3n + 3 = 51, this equation is solved to obtain n = 16, and verification confirms that 16 + 17 + 18 = 51 satisfies the problem conditions.

A. 15

This value would produce the consecutive triplet 15, 16, 17 summing to 48 falling 3 units short of the required 51.The error likely originates from solving 3n = 51 to get n = 17 (the middle integer) then incorrectly subtracting 2 instead of 1 to reach the smallest (17 - 2 = 15), revealing confusion about positional relationships within consecutive sequences. Students might recognize that 51 ÷ 3 = 17 represents the average (middle value) but then misapply offset calculations subtracting 2 rather than 1 to reach the minimum. Another plausible pathway involves setting up the equation correctly (n + n+1 + n+2 = 51 → 3n + 3 = 51) but then subtracting 3 from 51 to get 48 then dividing by 3 to obtain n = 16 correctly, yet mis-transcribing as 15 during answer selection. The consistent 3-unit deficit corresponds precisely to the sum of offsets (0 + 1 + 2 = 3) that must be accounted for when relating total sum to base integer highlighting how failure to properly handle the constant term in the equation produces systematic underestimation. This distractor effectively identifies learners with partial algebraic setup knowledge who recognize average relationships but lack precision in offset management or equation solving a transitional deficiency requiring explicit emphasis on complete equation resolution before answer extraction.

B. 16

This value correctly identifies the smallest integer through systematic algebraic representation and solution with comprehensive verification. Algebraic setup: let smallest integer = n; consecutive integers = n, n+1, n+2. Sum equation: n + (n+1) + (n+2) = 51 → 3n + 3 = 51. Equation solving: subtract 3 from both sides → 3n = 48; divide by 3 → n = 16. Verification pathway 1: direct summation 16 + 17 + 18 = 51 confirms total. Verification pathway 2: average method 51 ÷ 3 = 17 (middle integer); smallest = 17 - 1 = 16. Verification pathway 3: largest integer method largest = n+2 = 18; 18 - 2 = 16 smallest. This solution demonstrates comprehensive mastery of consecutive integer problem-solving including algebraic representation with appropriate variable selection, equation formation capturing sequential relationships, systematic solution maintaining equality balance, and multiple verification approaches leveraging arithmetic properties (average as middle value in odd-count consecutive sequences). These integrated competencies prove essential for number theory applications, sequence analysis, algorithmic problem-solving, and real-world scheduling where consecutive quantity relationships model inventory batches, production runs, or temporal sequences requiring precise boundary determination.

C. 17

This value represents the middle integer rather than the smallest, producing the triplet 17, 18, 19 summing to 54 exceeding the target by 3 units. The error likely originates from correctly calculating the average (51 ÷ 3 = 17) but then misidentifying this average as the smallest rather than the middle value in the consecutive sequence a critical conceptual error regarding positional relationships in arithmetic sequences with odd counts. Students might execute 51 ÷ 3 = 17 correctly through mental mathematics but fail to recognize that for three consecutive integers, the average always equals the median (middle value), not the minimum. Another plausible pathway involves solving 3n = 51 → n = 17 then forgetting to account for the +3 constant in the sum equation (3n + 3 = 51), effectively solving for the middle integer rather than the smallest. The consistent 3-unit excess corresponds precisely to the sum of positive offsets (+1 + +2 = +3) that would be added if starting from the middle rather than smallest integer diagnostically valuable for identifying learners who grasp average concepts but lack sequence positional awareness, a deficiency requiring explicit instruction on how averages relate to sequence positions based on count parity (odd versus even term counts).

D. 18

This figure represents the largest integer in the sequence rather than the smallest, yielding the triplet 18, 19, 20 summing to 57 exceeding the target by 6 units. The error likely originates from calculating the average (17) then adding 1 to reach 18 without recognizing this produces the largest rather than smallest value, or from solving 3n = 54 (incorrectly adding the offset sum to the total before division) then dividing to get n = 18. Students might execute 51 + 3 = 54 (correctly identifying the offset sum) but then divide 54 ÷ 3 = 18 without recognizing this yields the largest integer when offsets are +0, +1, +2 relative to smallest revealing confusion about reference points in offset calculations. Another plausible pathway involves working backward from an assumed largest value without systematic setup. The 6-unit excess (57 - 51 = 6) equals precisely twice the offset sum (2 × 3 = 6), suggesting systematic misalignment in reference point selection. This distractor effectively targets learners who recognize offset concepts but lack clarity about whether offsets are measured from smallest, middle, or largest reference highlighting the necessity of explicit variable definition ("let n = smallest") before equation setup to prevent reference ambiguity that cascades into solution errors with meaningful magnitude discrepancies.

Conclusion

The smallest integer 16 emerges through rigorous algebraic representation defining n as the minimum value, forming the sum equation capturing sequential offsets, solving systematically while maintaining equality balance, and verifying through multiple independent methods including direct summation and average-position relationships. This problem reinforces critical sequential reasoning competencies essential across mathematical domains: recognizing that consecutive integers differ by exactly 1, understanding how averages relate to sequence positions based on term count parity, executing algebraic setup with explicit variable definition to prevent reference ambiguity, and verifying solutions through reconstruction to detect procedural errors. Mastery of these integrated skills proves indispensable for number theory (divisibility patterns in sequences), computer science (array indexing, loop boundaries), operations research (batch sizing optimization), and temporal scheduling (consecutive day planning) where precise boundary determination prevents off-by-one errors with potentially severe consequences in algorithmic execution, resource allocation, or timeline management. The distractors strategically target reference point confusion (smallest versus middle versus largest) and offset miscalculation highlighting how seemingly minor definitional ambiguities produce systematic solution errors requiring explicit variable declaration protocols before equation formation to ensure consistent reference frameworks supporting accurate quantitative modeling of sequential relationships.