What is the least common multiple of 10 and 14?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 10 and 14. To solve this, we need to find the smallest number that is a multiple of both 10 and 14.
Answer
$70$
Answer for screen readers
The least common multiple (LCM) of 10 and 14 is $70$.
Steps to Solve
- Find the prime factorization of each number
To find the LCM, we first need to find the prime factors of each number.
For 10, the prime factorization is: $$ 10 = 2^1 \times 5^1 $$
For 14, the prime factorization is: $$ 14 = 2^1 \times 7^1 $$
- Identify the highest power of each prime factor
Next, we list all the prime factors from both numbers and take the highest power that appears in either factorization.
- For the prime factor 2, the highest power is $2^1$.
- For the prime factor 5, the highest power is $5^1$.
- For the prime factor 7, the highest power is $7^1$.
- Calculate the LCM using the prime factors
Now, we multiply the highest powers of all the prime factors together to find the LCM.
So the LCM is: $$ LCM = 2^1 \times 5^1 \times 7^1 $$
- Perform the multiplication
Now we will calculate the product: $$ LCM = 2 \times 5 \times 7 $$
Calculating this step by step: $$ 2 \times 5 = 10 $$ $$ 10 \times 7 = 70 $$
Thus, the least common multiple is 70.
The least common multiple (LCM) of 10 and 14 is $70$.
More Information
The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both. In this case, the LCM signifies a common quantity that can evenly distribute the quantities related to both numbers, useful in solving problems with fractions or common denominators.
Tips
- A common mistake is to confuse LCM with GCD (Greatest Common Divisor). Remember, LCM is about the smallest multiple, while GCD is about the largest factor.
- Another mistake is to overlook the prime factorizations, leading to incorrect calculations. Always check your factorization carefully!
