# Original derivation: prime factors, common divisors and common multiples

Author: /root/article_author. Written and actually checked 2026-10-05 (JST and UTC date).
Scope: positive-integer arithmetic and the three exact current exercise conditions. No external factual claim, teaching-effect claim, or real-world measurement is supported here. This is a new derivation, not a re-dated historical cache or a review summary.

## Definitions and direction
A positive divisor d of n satisfies n = d k for a positive integer k. A positive multiple m of n satisfies m = n k. A prime is an integer greater than one whose positive divisors are one and itself. In a prime factorization every factor must be prime. An exponent records repeated multiplication: 2² = 2 × 2, 3² = 3 × 3. A greatest common divisor is the largest positive integer dividing both given integers. A least common multiple is the smallest positive integer divisible by both.

## ma-0028: fully factoring 84
84 = 4 × 21 = 2 × 2 × 3 × 7 = 2² × 3 × 7. The displayed factors 2, 3 and 7 are prime: the only potential nontrivial divisor of 2 is absent; for 3 no integer lies strictly between 1 and 3 that divides it; for 7, checking 2 and 3 suffices because any factor pair of a composite 7 would have a member at most sqrt(7) < 3. These factors multiply to 84. Option a is 2 × 9 × 7 = 126, not 84; b is 8 × 7 = 56; c is 3 × 4 × 7 = 84 but 4 = 2 × 2 is not prime, so it is an incomplete factorization; d equals 84 and contains only primes. Thus d alone satisfies both the value and completeness conditions.

## ma-0029: greatest common divisor of 18 and 30
18 = 2 × 3², 30 = 2 × 3 × 5. Their prime counts are respectively (2:1,3:2,5:0) and (2:1,3:1,5:1). A divisor of both can use no more copies of each prime than either number supplies. The shared counts (2:1,3:1,5:0) give 6. For an independent finite maximality check, all positive divisors of 18 are 1,2,3,6,9,18: factor pairs 1×18,2×9,3×6 exhaust pairs by checking integers up to sqrt(18) < 5, with 4 failing divisibility. All divisors of 30 are 1,2,3,5,6,10,15,30: pairs 1×30,2×15,3×10,5×6 exhaust the integers up to sqrt(30) < 6, with 4 failing. Their intersection is 1,2,3,6, so 6 is greatest. Option a=6 divides both and is largest; b=3 divides both but is smaller; c=12 divides neither; d=90 is a multiple of both rather than a divisor. Only a is correct. Verification: 18÷6=3, 30÷6=5.

## ma-0030: least common multiple of 12 and 18
12 = 2² × 3, 18 = 2 × 3². The positive multiple sought must contain enough factors to include both sets: two copies of 2 and two copies of 3, giving 2² × 3² = 36. Check 36÷12=3 and 36÷18=2. For a separate finite minimality proof, the only positive multiples of 12 below 36 are 12 and 24; neither is divisible by 18. As every common multiple must first be a multiple of 12, none below 36 works. Of the given options, a=216 is divisible by both but exceeds 36; b=36 is common and minimal; c=6 is a common divisor and is not a multiple of either; d=72 is common but exceeds 36. Only b is correct. A prime exponent that is zero means the corresponding prime is absent, as 5 is absent from 18 in the previous example.

## Paired practice and pathway
The article practice uses only these three approved IDs without changing any condition, value, answer, explanation, source or diagram contract. Pencil-and-paper two-row prime-count notes and reverse division checks add a method rather than new bank items. The difference between asking what divides both and what is divided by both is the central new idea. Prerequisite: existing math-everyday in the same language, for interpreting division. Related and next reading: existing math-fractions-ratios in the same language, where common factors connect naturally to preserving a fraction's value. Prior articles cover quantities, fractions/ratios, geometry, summaries, basic probability, units, expressions, graphs, counting endpoints and conditional probability models; none teaches prime decomposition completeness and the common-divisor/common-multiple contrast. This article does not merely replace their numbers.

## Limits
Difficulty is the existing uncalibrated middle/standard classification of the referenced questions. No actual reader trial, improvement in learning, search ranking, traffic, revenue, or real Google aggregation is established. Source-bank historical records are retained below with their original dates; this proof does not pretend to re-fetch their curriculum links. Independent review must read this proof and the article, blind-solve the projected questions first, then inspect every current option and explanation. The writer does not grant approval.