What are the GCF and LCM of 8 and 12?

The first step to this method of finding the Least Common Multiple of 8 and 12 is to begin to list a few multiples for each number. If you need a refresher on how to find the multiples of these numbers, you can see the walkthroughs in the links below for each number.

Let’s take a look at the multiples for each of these numbers, 8 and 12:

What are the Multiples of 8?

What are the Multiples of 12?

Let’s take a look at the first 10 multiples for each of these numbers, 8 and 12:

First 10 Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80

First 10 Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120

You can continue to list out the multiples of these numbers as long as needed to find a match. Once you do find a match, or several matches, the smallest of these matches would be the Least Common Multiple. For instance, the first matching multiple(s) of 8 and 12 are 24, 48, 72. Because 24 is the smallest, it is the least common multiple.

The LCM of 8 and 12 is 24.

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Answer:

LCM:The first multiple that 8 and 12 have in common is 24. Notice that 48 is also a common multiple; however, 24 is the smallest number that they have in common. This makes it the least common multiple.

GCF:The GCF for 8 and 12 equals 4

The GCF for 8 and 12 equals 4The lcm or least common multiple is the smallest positive integer that two or more numbers will divide into evenly.

Least common multiple or lowest common denominator (lcd) can be calculated in two way; with the LCM formula calculation of greatest common factor (GCF), or multiplying the prime factors with the highest exponent factor.

Least common multiple (LCM) of 8 and 12 is 24.

LCM(8,12) = 24

Least Common Multiple of 8 and 12 with GCF Formula

The formula of LCM is LCM(a,b) = ( a × b) / GCF(a,b).
We need to calculate greatest common factor 8 and 12, than apply into the LCM equation.

GCF(8,12) = 4 LCM(8,12) = ( 8 × 12) / 4 LCM(8,12) = 96 / 4

LCM(8,12) = 24

Least Common Multiple (LCM) of 8 and 12 with Primes

Least common multiple can be found by multiplying the highest exponent prime factors of 8 and 12. First we will calculate the prime factors of 8 and 12.

Prime Factorization of 8

Prime factors of 8 are 2. Prime factorization of 8 in exponential form is:

8 = 23

Prime Factorization of 12

Prime factors of 12 are 2, 3. Prime factorization of 12 in exponential form is:

12 = 22 × 31

Now multiplying the highest exponent prime factors to calculate the LCM of 8 and 12.

LCM(8,12) = 23 × 31
LCM(8,12) = 24

LCM of 8 and 12 is 24. LCM also known as Least Common multiple or Lowest common multiple is the smallest or the least positive integer that is divisible by the given set of numbers. Consider the example for finding the LCM of 8 and 12. The answer is 24. 24 is divisible by both 8 and 12. Even 48 is divisible by 8 and 12, however it is not the LCM for 8 and 12. The smaller number than 48 is 24 which is divisible by both 8 and 12. Hence 24 is the Least Common Multiple for 8 and 12. You can refer to LCM for more information.

What is LCM of 8 and 12

The Least Common Multiple or Lowest Common Multiple of 8 and 12 is 24.

How to Find LCM of 8 and 12?

LCM of 8 and 12 can be determined using three methods:

Prime Factorisation
Division method
Listing the multiples

LCM of 8 and 12 Using Prime Factorisation Method

We can write 8 and 12 as the product of prime numbers, such that;

8 = 2 x 2 x 2

12 = 2 x 2 x 3

LCM (8, 12) = 2 x 2 x 2 x 3 = 24

LCM of 8 and 12 Using Division Method

In the Division Method, the given set of numbers are written in the same row separated by a comma. These numbers are divided with the smallest number that divides all, until no further division is possible or only when prime numbers are left.

2	8	12
2	4	6
2	2	3
3	1	3
x	1	1

LCM (8, 12) = 2 x 2 x 2 x 3 = 24

LCM of 8 and 12 Using Listing the Multiples

By listing all the multiples of given numbers, we can identify the first/smallest/least common multiple, which is the LCM. Below is the list of multiples for 8 and 12

Multiples of 8	Multiples of 12
8	12
16	24
24	36
32	48

LCM (8, 12) = 24

Video Lesson on Applications of LCM

Solved Examples

What is the smallest number that is divisible by both 8 and 12?

Answer: 24 is the smallest number that is divisible by both 8 and 12.

2. Two school buses arrive every 8 and 12 minutes at the school ground. At what time in minutes do both buses arrive together at the playground?

Answer: At the 24th minute, both the buses arrive together at the playground. This is a problem to find out the least common multiple of 8 and 12 and hence the answer is 24.

The LCM of 8 and 12 is 24.

No. LCM of 8 and 12 is 24 and LCM of 4 and 8 is 8. They are not the same.

Yes. The LCM (Least common multiple) and LCD (Least Common Divisor) are both the same for 8 and 12 which is 24.

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LCM of 8 and 12 is the smallest number among all common multiples of 8 and 12. The first few multiples of 8 and 12 are (8, 16, 24, 32, . . . ) and (12, 24, 36, 48, 60, . . . ) respectively. There are 3 commonly used methods to find LCM of 8 and 12 - by division method, by prime factorization, and by listing multiples.

What is the LCM of 8 and 12?

Answer: LCM of 8 and 12 is 24.

Explanation:

The LCM of two non-zero integers, x(8) and y(12), is the smallest positive integer m(24) that is divisible by both x(8) and y(12) without any remainder.

Methods to Find LCM of 8 and 12

Let's look at the different methods for finding the LCM of 8 and 12.

By Prime Factorization Method
By Division Method
By Listing Multiples

LCM of 8 and 12 by Prime Factorization

Prime factorization of 8 and 12 is (2 × 2 × 2) = 23 and (2 × 2 × 3) = 22 × 31 respectively. LCM of 8 and 12 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 23 × 31 = 24.
Hence, the LCM of 8 and 12 by prime factorization is 24.

LCM of 8 and 12 by Division Method

To calculate the LCM of 8 and 12 by the division method, we will divide the numbers(8, 12) by their prime factors (preferably common). The product of these divisors gives the LCM of 8 and 12.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 8 and 12. Write this prime number(2) on the left of the given numbers(8 and 12), separated as per the ladder arrangement.
Step 2: If any of the given numbers (8, 12) is a multiple of 2, divide it by 2 and write the quotient below it. Bring down any number that is not divisible by the prime number.
Step 3: Continue the steps until only 1s are left in the last row.

The LCM of 8 and 12 is the product of all prime numbers on the left, i.e. LCM(8, 12) by division method = 2 × 2 × 2 × 3 = 24.

LCM of 8 and 12 by Listing Multiples

To calculate the LCM of 8 and 12 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 8 (8, 16, 24, 32, . . . ) and 12 (12, 24, 36, 48, 60, . . . . )
Step 2: The common multiples from the multiples of 8 and 12 are 24, 48, . . .
Step 3: The smallest common multiple of 8 and 12 is 24.

∴ The least common multiple of 8 and 12 = 24.

☛ Also Check:

Example 1: Find the smallest number that is divisible by 8 and 12 exactly.

Solution:

The smallest number that is divisible by 8 and 12 exactly is their LCM.
⇒ Multiples of 8 and 12:
- Multiples of 8 = 8, 16, 24, 32, 40, 48, . . . .
- Multiples of 12 = 12, 24, 36, 48, 60, 72, . . . .
Therefore, the LCM of 8 and 12 is 24.
Example 2: Verify the relationship between GCF and LCM of 8 and 12.

Solution:

The relation between GCF and LCM of 8 and 12 is given as, LCM(8, 12) × GCF(8, 12) = Product of 8, 12
Prime factorization of 8 and 12 is given as, 8 = (2 × 2 × 2) = 23 and 12 = (2 × 2 × 3) = 22 × 31

LCM(8, 12) = 24 GCF(8, 12) = 4 LHS = LCM(8, 12) × GCF(8, 12) = 24 × 4 = 96 RHS = Product of 8, 12 = 8 × 12 = 96 ⇒ LHS = RHS = 96
Hence, verified.

Example 3: The GCD and LCM of two numbers are 4 and 24 respectively. If one number is 12, find the other number.

Solution:

Let the other number be z.
∵ GCD × LCM = 12 × z ⇒ z = (GCD × LCM)/12 ⇒ z = (4 × 24)/12 ⇒ z = 8

Therefore, the other number is 8.

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The LCM of 8 and 12 is 24. To find the least common multiple (LCM) of 8 and 12, we need to find the multiples of 8 and 12 (multiples of 8 = 8, 16, 24, 32; multiples of 12 = 12, 24, 36, 48) and choose the smallest multiple that is exactly divisible by 8 and 12, i.e., 24.

What is the Least Perfect Square Divisible by 8 and 12?

The least number divisible by 8 and 12 = LCM(8, 12)
LCM of 8 and 12 = 2 × 2 × 2 × 3 [Incomplete pair(s): 2, 3]
⇒ Least perfect square divisible by each 8 and 12 = LCM(8, 12) × 2 × 3 = 144 [Square root of 144 = √144 = ±12]
Therefore, 144 is the required number.