What Is the Lcm of 12 and 7?

The least common multiple (LCM) of two integers is the smallest positive integer that is a multiple of both numbers. For example, the LCM of 10 and 15 is 30 because 30 is the smallest positive integer that is a multiple of both 10 and 15.

To find the LCM of two numbers, you can use the greatest common divisor (GCD) method. This method is also known as the highest common factor (HCF) method.

The GCD of two numbers is the largest positive integer that divide both numbers without leaving a remainder. For example, the GCD of 10 and 15 is 5 because 5 is the largest positive integer that divides both 10 and 15 without leaving a remainder.

To find the LCM of 12 and 7 using the GCD method, you first need to find the GCD of the two numbers. To do this, you can use the Euclidean algorithm.

The Euclidean algorithm is a method for finding the GCD of two numbers. It is named after the Greek mathematician Euclid, who first described it in his book The Elements.

The Euclidean algorithm is based on the following property:

"If a and b are two positive integers, then the GCD of a and b is equal to the GCD of a-b and b."

For example, let's say we want to find the GCD of 10 and 15. We can use the Euclidean algorithm to do this.

First, we subtract 10 from 15. This gives us a result of 5.

Next, we subtract 5 from 10. This gives us a result of 5.

Finally, we subtract 5 from 5. This gives us a result of 0.

The last number we subtract is the GCD of the two numbers. In this case, the GCD of 10 and 15 is 5.

Now that we know the GCD of the two numbers, we can use it to find the LCM.

To find the LCM of 12 and 7, we use the following formula:

LCM(12,7) = (12 * 7) / GCD(12,7)

LCM(12,7) = (12 * 7) / 5

LCM(12,7) = 84 / 5

LCM(12,7) = 16

Intriguing read: 7 15

The greatest common divisor of 12 and 7 is 1.

The highest common factor of 12 and 7 is 1.

The smallest number that is a multiple of both 12 and 7 is 84.

The largest number that is a multiple of both 12 and 7 is 84.

The smallest positive integer that is divisible by both 12 and 7 is 84. The number 84 is divisible by 12 because it is evenly divisible by 3 (12 = 3 × 4), and it is divisible by 7 because it is evenly divisible by 7. Thus, we can say that 84 is the smallest positive integer that is divisible by both 12 and 7.

The number 84 is an important number in many respects. For example, it is the product of the first three odd prime numbers (3 × 7 × 11), and it is the sum of the squares of the first four integers (1^2 + 2^2 + 3^2 + 4^2 = 84).

In addition, the number 84 is significant in the field of mathematics known as number theory. In particular, it is a so-called "happy number" (a number that leads to 1 whenrepeatedly summed with the squares of its digits), and it is also a "highly composite number" (a number with more divisors than any smaller number).

For another approach, see: What Is the Lcm of 4 and 14?

The largest positive integer that is divisible by both 12 and 7 is 84.

Since both 12 and 7 are divisors of 84, this means that 84 is evenly divisible by both of them. You can use the division algorithm to check this:

12 goes into 84 seven times, with a remainder of 0.

7 goes into 84 twelve times, also with a remainder of 0.

That means that 84 is divisible by both 12 and 7.

Interestingly, 84 is also divisible by their least common multiple, which is 21. This is because the LCM of 12 and 7 is 21, and 84 is divisible by 21 (it goes into 84 four times, with a remainder of 0).

So, the largest positive integer that is divisible by both 12 and 7 is 84.

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both of the numbers. To find the LCM of 12 and 7, we can list the first several multiples of each number and find the first number that appears in both lists:

Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96 Multiples of 7: 7, 14, 21, 28, 35, 42, 49

Thus, the LCM of 12 and 7 is 49.

It is sometimes helpful to think of the LCM as the smallest number that both numbers will go into evenly. For example, we can say that the LCM of 12 and 7 is 49 because:

12 goes into 49 4 times (49 ÷ 12 = 4) 7 goes into 49 7 times (49 ÷ 7 = 7)

This is why the LCM is also sometimes called the smallest common multiple or smallest common factor.

Explore further: What Is the Lcm of 4 and 10?

The greatest common factor of 12 and 7 is 1.