Factors, Multiples, GCF & LCM

Factors and multiples

A factor is a number that divides evenly into another number, leaving no remainder. The factors of 12 are 1, 2, 3, 4, 6, and 12, because each one divides 12 exactly.

A multiple is what you get when you skip-count by a number. The multiples of 4 are 4, 8, 12, 16, 20, and so on. Notice that every number is a factor of its own multiples.

A quick way to test factors: divide the number by each candidate. If the quotient is a whole number with no remainder, you found a factor.

Every whole number has at least two factors: 1 and itself. Prime numbers have exactly those two, which is why primes are the building blocks for bigger numbers.

Greatest Common Factor (GCF)

The GCF of two numbers is the largest factor they share. List the factors of each number, then pick the biggest one that appears in both lists.

GCF is useful when you need the largest equal-sized groups you can make from two quantities. Example: 18 pencils and 24 erasers shared into identical kits with nothing left over. The kit size is the GCF.

Another use: simplifying fractions. The GCF of the numerator and denominator is the biggest number you can divide both by at once.

Least Common Multiple (LCM)

The LCM is the smallest multiple that two numbers share. Skip-count by each number until you reach the first value they both land on.

LCM answers questions like: "When will both buses arrive at the same time again?" or "What is the smallest batch size that works for two different recipes?"

If two denominators are different, the LCM of those denominators is often the least common denominator you need for adding fractions.

Why this matters

Numbers that divide evenly, numbers you count by, and the values they share.

These ideas show up on homework, quizzes, and the next unit in this grade. Getting comfortable here saves time later when the problems stack more steps together.

Common mistakes to avoid

Picking GCF when the problem asks for LCM, or the other way around. Check whether you need the biggest shared factor or the next time both counts line up.

Assuming the LCM is always the product of the two numbers. That only works when the numbers share no factors besides 1.

Skipping 1 when you list factors. Every whole number has 1 as a factor.

Key ideas from this lesson

1. Factors and multiples
2. Greatest Common Factor (GCF)
3. Least Common Multiple (LCM)