Solving Word Problems with the GCF and LCM | Grade 7 Math | LevelKite

Some word problems are really questions about splitting amounts into equal groups, or about repeating amounts until they line up. In this lesson you learn to tell which kind of story you are reading and how to solve it.

Here are the words you need. Every quantity in the stories is a positive whole number ( $1, 2, 3$ and so on), because you count whole objects, whole minutes or whole centimeters.

A factor of a number is a whole number that divides it with no remainder. For example, $4$ is a factor of $12$ because $12 \div 4 = 3$ .
A multiple of a number is that number times a positive whole number. For example, $36$ is a multiple of $12$ because $12 \times 3 = 36$ .
The greatest common factor (GCF) of two numbers is the largest number that is a factor of both.
The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both.

You already know how to find a GCF and an LCM. The new skill is choosing the right one for a story.

Splitting stories. Suppose you share $12$ stickers and $18$ cards into identical bags, so every bag gets the same number of stickers and the same number of cards, with nothing left over. Could you use $6$ bags? Yes: each bag gets $12 \div 6 = 2$ stickers and $18 \div 6 = 3$ cards. Could you use $5$ bags? No: $12 \div 5$ is not a whole number, so you cannot share the stickers equally.

The number of bags works only when it divides both $12$ and $18$ with no remainder. A number that is a factor of both is a common factor. The common factors of $12$ and $18$ are $1$ , $2$ , $3$ and $6$ , so those are the only possible numbers of bags. The greatest number of bags is the greatest common factor, $6$ .

Matching-up stories. Now suppose one light blinks every $4$ seconds and another blinks every $6$ seconds, and they blink together at the start. Count the seconds from that start. The first light blinks at $4, 8, 12, 16, \dots$ seconds, which are the multiples of $4$ . The second light blinks at $6, 12, 18, \dots$ seconds, which are the multiples of $6$ .

The lights blink together only at a time that is in both lists. A number that is a multiple of both is a common multiple. The common multiples of $4$ and $6$ are $12, 24, 36, \dots$ , so the lights next blink together at the least common multiple, $12$ seconds.

Key concept

Does the unknown divide, or is it divided?

To choose between the GCF and the LCM, name the unknown, which is the number the story makes you find, and ask how it relates to the given numbers. This test applies when every quantity is a positive whole number.

If the unknown must divide every given number with no remainder, it is a common factor. When the story wants the largest one, find the GCF.
If every given number must divide the unknown with no remainder, it is a common multiple. When the story wants the smallest one, find the LCM.

The GCF or LCM is sometimes one step before the final answer, so always go back and answer the question the story asks.

Two quick checks for a GCF or an LCM of positive whole numbers

Check	If you found a GCF	If you found an LCM
Size check	It is never greater than the smallest given number.	It is never less than the largest given number.
Why the size check works	A factor of a number is never greater than that number.	A multiple of a number is never less than that number.
Division check	Each given number divided by the GCF is a whole number.	The LCM divided by each given number is a whole number.
Why the division check works	The GCF is a factor of every given number.	The LCM is a multiple of every given number.

Worked example

Splitting: identical supply kits

A teacher has $48$ pencils and $36$ erasers. She packs all of them into identical kits, so every kit has the same number of pencils and the same number of erasers. What is the greatest number of kits she can make, and what goes in each kit?

Name the unknown. Let $k$ be the number of kits, a positive whole number. Each kit gets $48 \div k$ pencils, and that must be a whole number because the kits share the pencils equally with none left over. In the same way, each kit gets $36 \div k$ erasers, which must also be a whole number.
$48 \div k \text{ and } 36 \div k \text{ are whole numbers}$
Choose the tool. Since $k$ divides both $48$ and $36$ , it is a common factor of $48$ and $36$ . The story asks for the greatest number of kits, so $k$ is the greatest common factor.
$k = \text{GCF}(48, 36)$
Find the GCF by listing factors. The factors of $48$ are $1, 2, 3, 4, 6, 8, 12, 16, 24, 48$ . The factors of $36$ are $1, 2, 3, 4, 6, 9, 12, 18, 36$ . The numbers in both lists are $1, 2, 3, 4, 6, 12$ , and the largest of them is $12$ .
$\text{GCF}(48, 36) = 12$
Answer the second question. Each kit gets an equal share, so divide each amount by the $12$ kits.
$48 \div 12 = 4 \qquad 36 \div 12 = 3$
Check. The size check passes because $12$ is not greater than $36$ , the smallest given number. The division check passes because both results in the last step are whole numbers. So $12$ is a common factor, and the lists show it is the greatest. Multiplying back confirms that $12$ kits use every item.
$12 \times 4 = 48 \qquad 12 \times 3 = 36$

Answer

The greatest number of kits is $12$ , and each kit holds $4$ pencils and $3$ erasers.

Worked example

Matching up: two bus routes

Route A buses leave a station every $12$ minutes and Route B buses leave every $18$ minutes. One bus from each route leaves at 9:00 a.m. What is the next time that buses from both routes leave together?

Name the unknown. Count minutes from 9:00 a.m. Both routes start then and keep a fixed gap, so Route A leaves at $12, 24, 36, \dots$ minutes and Route B at $18, 36, 54, \dots$ minutes. Let $t$ be the number of minutes until both next leave together, a positive whole number. Then $t$ is a multiple of $12$ and a multiple of $18$ .
$t \div 12 \text{ and } t \div 18 \text{ are whole numbers}$
Choose the tool. Since $12$ and $18$ both divide $t$ , it is a common multiple of $12$ and $18$ . The story asks for the next time, which is the smallest such $t$ , so $t$ is the least common multiple.
$t = \text{LCM}(12, 18)$
Find the LCM by listing multiples until a number appears in both lists. The multiples of $12$ are $12, 24, 36$ and so on. The multiples of $18$ are $18, 36$ and so on. The first number in both lists is $36$ .
$\text{LCM}(12, 18) = 36$
Check. The size check passes because $36$ is not less than $18$ , the largest given number. The division check passes because both results are whole numbers: $36$ minutes is the 3rd Route A departure and the 2nd Route B departure after 9:00 a.m. So $36$ is a common multiple, and the lists show it is the least.
$36 \div 12 = 3 \qquad 36 \div 18 = 2$
Answer the question. It asks for a clock time, so add $36$ minutes to 9:00 a.m.
$\text{9:00 a.m.} + 36 \text{ min} = \text{9:36 a.m.}$

Answer

Buses from both routes next leave together at 9:36 a.m., which is $36$ minutes after 9:00 a.m.

Worked example

One more step: packs to buy

Juice boxes come in packs of $8$ and granola bars come in packs of $12$ . You want to buy both, with exactly as many juice boxes as granola bars, using as few packs as possible. How many packs of each should you buy?

Let $n$ be the number of juice boxes, which is also the number of granola bars. It is a positive whole number. Whole packs give $8, 16, 24, \dots$ juice boxes and $12, 24, 36, \dots$ granola bars, so $8$ and $12$ both divide $n$ . That makes $n$ a common multiple of $8$ and $12$ .
$n \div 8 \text{ and } n \div 12 \text{ are whole numbers}$
Every extra pack adds items, so the fewest packs go with the smallest $n$ . That is the least common multiple. In the two lists from the first step, the first shared number is $24$ .
$n = \text{LCM}(8, 12) = 24$
The question asks for packs, not items, so divide $24$ by each pack size. Both results are whole numbers, so the division check passes as well.
$24 \div 8 = 3 \qquad 24 \div 12 = 2$

Answer

You should buy $3$ packs of juice boxes and $2$ packs of granola bars, which gives $24$ of each.

Takeaway. In a word problem, the story chooses the tool; a single word does not. Name the unknown and ask how it relates to the given positive whole numbers. If it must divide each of them, it is a common factor, and the largest one is the GCF. If each of them must divide it, it is a common multiple, and the smallest one is the LCM. Then check your result and return to the question, because the GCF or LCM may be one step before the answer the story asks for.