2.3 Properties of Operations, Factors, GCF & LCM

2.3 Properties of Operations, Factors, GCF & LCM

Competency 1 pairs two skills that look unrelated but serve the same purpose: they are the tools that make multi-step arithmetic manageable. Skill 7 covers the properties of operations that license rearrangement; skill 5 covers factor structure — common factors, greatest common factor, and least common denominator.

Naming the properties precisely

Identification items give an equation and ask which property justifies it. The distinctions are exact.

The tell for commutative is that the order of terms changed. The tell for associative is that the grouping changed while the order stayed the same. Items exploit this by showing (3 + 8) + 5 = 3 + (8 + 5) — order unchanged, grouping changed, so associative — next to an option labelled commutative.

[!NOTE] Subtraction and division are neither commutative nor associative. 12 − 5 = 7 but 5 − 12 = −7. And (24 ÷ 6) ÷ 2 = 2 while 24 ÷ (6 ÷ 2) = 8. Middle-grades students routinely over-generalize the addition properties to subtraction; recognizing that error is also a Competency 5 task.

The distributive property is the one that does real computational work. It is why 6 · 47 can be computed mentally as 6(50 − 3) = 300 − 18 = 282, and it is the justification behind combining like terms, expanding binomials, and factoring.

Prime factorization

Every integer greater than 1 factors uniquely into primes — the Fundamental Theorem of Arithmetic. Build the factorization by dividing repeatedly by the smallest prime that divides:

• 180 = 2 · 90 = 2 · 2 · 45 = 2² · 3² · 5
• 168 = 2 · 84 = 2² · 42 = 2³ · 21 = 2³ · 3 · 7

Divisibility shortcuts speed this up: a number is divisible by 2 if it ends in an even digit; by 3 if its digit sum is divisible by 3; by 4 if its last two digits form a multiple of 4; by 5 if it ends in 0 or 5; by 6 if it passes both the 2 and 3 tests; by 9 if its digit sum is divisible by 9; by 10 if it ends in 0.

GCF and LCM from prime factorization

+---------------------------------------------------------------------------+
|            180 = 2^2 * 3^2 * 5          168 = 2^3 * 3 * 7                 |
+---------------------------------------------------------------------------+
|  GCF: take each SHARED prime at its LOWEST power                          |
|       shared primes are 2 and 3  ->  2^2 * 3^1 = 12                       |
|                                                                           |
|  LCM: take EVERY prime at its HIGHEST power                               |
|       2^3 * 3^2 * 5 * 7 = 8 * 9 * 5 * 7 = 2520                            |
+---------------------------------------------------------------------------+
|  Check: GCF * LCM = 12 * 2520 = 30240,  and  180 * 168 = 30240            |
+---------------------------------------------------------------------------+

That identity — GCF(a, b) · LCM(a, b) = a · b — is worth memorizing. If an item tells you two numbers and their GCF and asks for the LCM, you can divide rather than factor.

The least common denominator of two fractions is simply the LCM of their denominators. To compute 7/12 − 3/20: 12 = 2² · 3 and 20 = 2² · 5, so the LCM is 2² · 3 · 5 = 60. Then 35/60 − 9/60 = 26/60 = 13/30.

Deciding which one a word problem wants

This is the single most reliable source of GCF/LCM items, and there is a clean decision rule.

Use the GCF when you are breaking something apart into equal groups and want the largest possible group size or the greatest number of identical bundles. Signals: "greatest number of," "largest possible," "without any left over," "identical kits."

A teacher has 84 protractors and 126 rulers and wants to build identical kits using all the items. The greatest number of kits is GCF(84, 126). With 84 = 2² · 3 · 7 and 126 = 2 · 3² · 7, the GCF is 2 · 3 · 7 = 42 kits, each holding 2 protractors and 3 rulers.

Use the LCM when two repeating cycles must line up, or when you need a common denominator. Signals: "at the same time again," "next time both," "smallest number that is a multiple of."

One bell rings every 12 minutes and another every 18 minutes. They ring together again after LCM(12, 18) minutes. Since 12 = 2² · 3 and 18 = 2 · 3², the LCM is 2² · 3² = 36 minutes.

The distractor pattern is predictable: the kit problem offers 252 (the LCM) and the bell problem offers 6 (the GCF). Read for whether the quantities are being divided up or repeated forward.

Using factor structure to simplify computation

Factor knowledge shortens arithmetic you would otherwise hand to a slow on-screen calculator:

• Reducing fractions: 231/385 looks hostile until you see 231 = 3 · 7 · 11 and 385 = 5 · 7 · 11, so it reduces to 3/5.
• Cancelling before multiplying: (18/35)(25/27) becomes (18/27)(25/35) = (2/3)(5/7) = 10/21.
• Regrouping for mental math: 25 · 17 · 4 is best done as (25 · 4) · 17 = 100 · 17 = 1700, an application of the commutative and associative properties together.