Find greatest common factors and least common multiples

Learning goal: Find the greatest common factor of two whole numbers at most one hundred and the least common multiple of two positive whole numbers at most twelve. Use factor structure and the distributive property to rewrite a sum as a common factor times a sum whose terms share no factor greater than one.

A factor divides a whole number without a remainder. The positive factors of eighteen are 1, 2, 3, 6, 9 and 18. Those of thirty are 1, 2, 3, 5, 6, 10, 15 and 30. Their common factors are 1, 2, 3 and 6, so the greatest common factor, or GCF, is six. Greatest means the largest shared factor, not the larger original number.

Suppose eighteen blue cards and thirty gold cards are distributed into the greatest possible number of identical packs with no cards left. Each pack must contain the same number of blue cards and the same number of gold cards. The pack count must divide both eighteen and thirty. Six packs work, with three blue and five gold in each. Nine packs fail because thirty cannot be divided into nine equal whole-card groups.

The diagram shows six actual packs, each containing three blue and five gold cards. Counting all packs recovers eighteen blue and thirty gold. The common factor six counts packs; the numbers three and five count cards of each color per pack. The total card count is forty-eight.

The distributive property expresses that grouping: 18 + 30 = 6 × 3 + 6 × 5 = 6(3 + 5). Three and five share no positive common factor greater than one. Factoring out only two gives 2(9 + 15), which is equivalent but leaves another common factor inside. Factoring out the GCF leaves inner terms with no common factor greater than one, which is the form the standard asks for.

A multiple is the result of multiplying a number by a whole number. Positive multiples of six include 6, 12, 18, 24 and 30. Positive multiples of eight include 8, 16, 24 and 32. The least common multiple, or LCM, is twenty-four, the smallest positive value in both lists. Zero is a multiple of both, but it is not the least positive common multiple sought here.

If two classroom signals begin together and repeat every six minutes and eight minutes, they next occur together after twenty-four minutes, assuming both schedules continue exactly. One signal occurs at 6, 12, 18 and 24 minutes; the other at 8, 16 and 24. The LCM answers the shared-timing question. The GCF, two, answers a different divisibility question and is not the next meeting time.

Prime factors provide another method. Eighteen is 2 × 3 × 3; thirty is 2 × 3 × 5. Their shared prime factors give 2 × 3 = 6 for the GCF. For six and eight, the factorizations are 2 × 3 and 2 × 2 × 2. A common multiple must include enough factors for each number, so the LCM is 2 × 2 × 2 × 3 = 24.

Multiplying the two numbers always gives a common multiple, but it may not give the least one. Six times eight is forty-eight, a shared multiple larger than twenty-four. Likewise, one is always a positive common factor, but it may not be the greatest. The words least and greatest require comparing eligible values or using factor structure that shows no greater common factor, or no smaller common multiple, exists.

When one number divides the other, the relationship is especially clear. For four and twelve, the GCF is four and the LCM is twelve. For relatively prime positive numbers such as five and eight, the GCF is one and the LCM is forty. Relatively prime describes the pair's shared factors, not a claim that both numbers individually are prime.

Worked distribution: For 36 + 84, the GCF is twelve. Write 12(3 + 7) = 120. The inner terms three and seven have no common factor greater than one. The equality can be checked by distributing twelve back to each term.

Worked choice of operation: Identical packs with no leftover objects require a common divisor of the two counts. Repeating schedules meeting again require a common multiple of the intervals. Decide what the unknown counts before choosing GCF or LCM.

Guided factor work: List factors of eighteen and thirty, then verify the six-pack model. List positive multiples of six and eight until the first shared value appears. Connect both lists with prime-factor reasoning and explain the different roles of the answers.

Independent performance: Find GCFs for pairs at most one hundred, LCMs for pairs at most twelve, and rewrite two sums using the GCF and distribution. Include one pair where one number divides the other and one relatively prime pair. Create a packing problem and a schedule problem, identify the required structure and justify why your answer is greatest or least as requested.