6th Grade · 6.NS.B.4

Greatest common factor in 6th grade

The greatest common factor of two numbers is the biggest number that divides both of them with nothing left over. In 6th grade your child finds it, along with its partner, the least common multiple. Your child also uses the greatest common factor to rewrite a sum such as $15 + 35$ as $5 \times (3 + 7)$ .

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.NS.B.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.

What it means, in plain words

A factor is a number that divides another number with nothing left over. The factors of 12 are 1, 2, 3, 4, 6 and 12. The greatest common factor (GCF) of two numbers is the biggest factor they share. For 12 and 18 it is 6. In 6th grade, children find the GCF of two whole numbers up to 100.

A multiple is what you get when you count by a number. The multiples of 4 are 4, 8, 12, 16, 20 and so on. The least common multiple (LCM) is the smallest multiple two numbers share. Counting by 10 gives 10, 20, so the LCM of 4 and 10 is 20. Children find the LCM of two numbers up to 12. GCF and LCM sound alike and get mixed up. The GCF is never bigger than the smaller number, and the LCM is never smaller than the bigger number.

Children also use the GCF to rewrite a sum with the distributive property, the rule that $5 \times (3 + 7)$ equals $5 \times 3 + 5 \times 7$ . Working backward, $15 + 35$ becomes $5 \times (3 + 7)$ : the common factor 5 comes out, and 3 and 7 share no factor except 1. This is an early step toward factoring in algebra.

Your child should be able to:

List the factors of two numbers up to 100 and pick out the greatest common factor
Count by two numbers up to 12 and find the least common multiple
Decide whether a word problem asks for the GCF or the LCM
Rewrite a sum such as $15 + 35$ as $5 \times (3 + 7)$

Example

What is the greatest common factor of 24 and 36, and how can you use it to rewrite $24 + 36$ ?

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
The biggest number on both lists is 12, so the GCF is 12.
$24 = 12 \times 2$ and $36 = 12 \times 3$ , so $24 + 36 = 12 \times (2 + 3)$ . Both sides equal 60.

Example

Hot dogs come in packs of 10 and buns come in packs of 8. What is the smallest number of each you can buy so that every hot dog has a bun?

Count by 10: 10, 20, 30, 40, 50.
Count by 8: 8, 16, 24, 32, 40.
The first number on both lists is 40, so the LCM is 40.
Buy 40 of each: 4 packs of hot dogs and 5 packs of buns.

Where children usually get stuck

Asked for the greatest common factor of 24 and 36, your child answers 6.
6 is a common factor, and the child stops at the first shared factor that comes to mind. The greatest one, 12, never gets checked.
What helps. Have your child list factors in pairs, such as 1 and 24, 2 and 12, 3 and 8, 4 and 6. Pairs make it hard to miss one. Then circle the numbers on both lists and pick the biggest.
For the hot dogs in packs of 10 and buns in packs of 8, your child finds the GCF, 2.
Both names have the word "common" and both use the same two numbers. The child does not connect the question to counting up or to splitting into groups.
What helps. Ask whether the answer will be bigger or smaller than the numbers in the problem. Buying packs until the counts match means counting up, so it is a multiple. Splitting into equal groups means a factor.
For the least common multiple of 6 and 9, your child multiplies $6 \times 9 = 54$ .
Multiplying two numbers always gives a common multiple, so the shortcut seems to work. It gives the least one only when the numbers share no factor except 1, and 6 and 9 share 3. The LCM is 18.
What helps. Have your child count by the bigger number (9, 18, 27) and stop at the first one the smaller number divides. 6 divides 18, so the LCM is 18.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Use the area model to factor the sum shown by taking out the GCF.
Type the missing side lengths: the shared side on the left and the two widths across the top.
6
30
Show the answer and how to solve it
The left side is 6, and the numbers across the top are 1 and 5, so $6 + 30$ factors as $6(1 + 5)$ .
1. 1Find the GCF of 6 and 30. Both are divisible by 6, and 6 is the largest such number, so the GCF is 6. Place 6 on the left side of the model.
2. 2Divide each cell area by 6 to find the widths across the top.
3. 3The top numbers 1 and 5 share no common factor greater than 1, so the factorization is complete.
Common mistake. A common mistake is placing 2 or 3 on the left because both divide 6 and 30. Using 3 gives top numbers 2 and 10, but those still share a factor of 2, so 3 is not the GCF.
The idea underneath. The shared side of both rectangles is the factor taken out of the sum; if the two top numbers still share a common factor greater than 1, you have not pulled out the GCF yet.
2On level
Look at the multiplication table below. Two rows are highlighted.
1
2
3
4
5
6
7
8
9
10
11
12
1
1
2
3
4
5
6
7
8
9
10
11
12
2
2
4
6
8
10
12
14
16
18
20
22
24
3
3
6
9
12
15
18
21
24
27
30
33
36
4
4
8
12
16
20
24
28
32
36
40
44
48
5
5
10
15
20
25
30
35
40
45
50
55
60
6
6
12
18
24
30
36
42
48
54
60
66
72
7
7
14
21
28
35
42
49
56
63
70
77
84
8
8
16
24
32
40
48
56
64
72
80
88
96
9
9
18
27
36
45
54
63
72
81
90
99
108
10
10
20
30
40
50
60
70
80
90
100
110
120
11
11
22
33
44
55
66
77
88
99
110
121
132
12
12
24
36
48
60
72
84
96
108
120
132
144
Enter the smallest product that appears in both highlighted rows.
Show the answer and how to solve it
Answer. 24
The answer is 24 because 24 is the first product that appears in both row 6 and row 8.
1. 1The highlighted rows are row 6 and row 8, so you are comparing multiples of 6 and multiples of 8.
2. 2Read across each row from left to right and list the products until you find a match.
3. 3The first number that appears in both lists is 24. That means 24 is the least common multiple.
Common mistake. A common mistake is multiplying 6 and 8 to get 48. While 48 does appear in both rows, 24 appears earlier, so 24 is the least common multiple.
The idea underneath. The least common multiple is the first number shared by two ordered lists of multiples.
3On level
What is the least common multiple of 4 and 6? Enter your answer as a whole number.
LCM:
Show the answer and how to solve it
Answer. 12
The least common multiple of 4 and 6 is 12 because 12 is the smallest number that is a multiple of both 4 and 6.
1. 1Write a few multiples of 4 and a few multiples of 6.
2. 2Look for the first number that appears in both lists. That number is 12.
3. 3Since 12 is the smallest shared multiple, it is the least common multiple.
Common mistake. A common mistake is multiplying 4 and 6 to get 24. That is a common multiple, but it is not the least one because 12 comes first and works for both numbers.
The idea underneath. The least common multiple is the first number that appears in both lists of multiples.

4Stretch

A student wrote the work shown below to find the LCM of 7 and 9.

\displaystyle \text{Multiples of 7: } 7, 14, 21, 28, 35, 42, 49, 56, 63

\displaystyle \text{Multiples of 9: } 9, 18, 27, 36, 45, 54, 63

Claim: "The LCM of 7 and 9 is 63."

Select the statement that best evaluates the student's claim.

The claim is correct.

The student used a common factor instead of a common multiple.

The number is a multiple of only one of the two numbers.

The number is a common multiple, but there is a smaller common multiple.

Show the answer and how to solve it

The claim is correct. The only common factor of 7 and 9 is 1, so their LCM equals their product, 63. It is the first shared value in both lists.

1The student listed multiples of 7 and multiples of 9, then claimed the LCM is 63.
2Check whether 63 is a common multiple. Both 7 and 9 divide evenly into 63, and 63 appears in both lists.
3Check whether 63 is the least. Because $\gcd(7, 9) = 1$ , the two numbers share no common factor other than 1, so the LCM must equal their product: $7 \times 9 = 63$ . No smaller common multiple can exist.
4The claim is correct. The best evaluation is: "The claim is correct."

Common mistake. A common mistake is assuming the product of two numbers is always too large to be the LCM. For relatively prime pairs like 7 and 9, the product is exactly the LCM—no smaller shared multiple exists.

The idea underneath. To evaluate an LCM claim, check two things: the number must be a multiple of both given numbers, and no smaller positive common multiple can come first.