Factors and Multiples
Lessons in this chapter
About Factors and Multiples
Factors and Multiples revisits 4th grade’s structure with one major addition: prime factorization, which is the tool that makes GCF and LCM mechanical instead of guesswork.
Apply Divisibility Rules for 2, 3, 5, 6, 9, and 10 and Apply Divisibility Rules for 4 and 8 extend the tests. The rules for 4 and 8 are new: a number is divisible by 4 if its last two digits are, and by 8 if its last three are, both of which follow from 100 and 1,000 being divisible by them.
Find the Prime Factorization of Numbers is the central lesson. Every whole number breaks into a unique product of primes, so 60 is 2 × 2 × 3 × 5, and no other combination of primes gives 60. That uniqueness is what makes the method reliable.
Find Greatest Common Factor (GCF) and Find Least Common Multiple (LCM) then stop being searches. The GCF is the primes the two numbers share, the LCM is every prime each number needs, and both are read straight off the factorizations.
Solve Applied Problems with GCF and LCM gives them purpose. GCF problems split things into equal groups as large as possible; LCM problems find when two repeating cycles coincide, like two buses leaving together again.
Every one of the six lessons in this chapter opens as a 6th grade math worksheet in the page. Find the Prime Factorization of Numbers, Find Greatest Common Factor (GCF) and Find Least Common Multiple (LCM) each come with math problems worked one at a time and scored as you go, which is the quickest way to tell whether the method has stuck or only the worked example has. It is free to try, with no login needed for your first 3.
Frequently asked questions
It opens with two lessons of divisibility rules, the familiar tests for 2, 3, 5, 6, 9 and 10 and then the less familiar ones for 4 and 8. Find the Prime Factorization of Numbers is the lesson the rest leans on, because once a number is broken into its primes, Find Greatest Common Factor (GCF) and Find Least Common Multiple (LCM) stop being searches and become calculations. Solve Applied Problems with GCF and LCM closes the chapter by separating the situations each one answers. It maps onto the Common Core domain The Number System (6.NS), where greatest common factor and least common multiple sit; the divisibility rules and prime factorization are the groundwork we teach to get there.
Breaking a number into a product of primes. 60 is 2 × 2 × 3 × 5, and every whole number has exactly one such breakdown.
Take the primes both numbers share, with the lowest power of each, and multiply them. It turns the GCF from a search into a calculation.
Take every prime that appears in either number, with the highest power of each, and multiply. The result is the smallest number both divide into.
If the last two digits form a number divisible by 4, the whole number is. It works because 100 is divisible by 4, so everything above the tens place already is.
GCF for splitting into the largest possible equal groups. LCM for finding when two repeating events coincide, like two buses that leave together again.