lcm of 14 and 42
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 14 and 42. To find the LCM, we typically determine the multiples of each number and identify the smallest multiple that is common to both, or we can use prime factorization or the LCM formula based on their greatest common divisor (GCD).
Answer
The least common multiple of 14 and 42 is \( 42 \).
Answer for screen readers
The least common multiple of 14 and 42 is ( 42 ).
Steps to Solve
- Determine the Prime Factorizations First, we need to find the prime factorization of both numbers.
For 14:
- 14 can be divided by 2 (which is prime): $14 = 2 \times 7$.
For 42:
- 42 can be divided by 2 (which is prime): $42 = 2 \times 21$.
- 21 can be further divided by 3 (which is prime): $21 = 3 \times 7$. Thus, $42 = 2 \times 3 \times 7$.
- Identify the Unique Prime Factors List the unique prime factors from both factorizations:
- From 14: 2, 7
- From 42: 2, 3, 7
The unique prime factors are: 2, 3, and 7.
- Use Each Prime Factor the Maximum Number of Times Found For the LCM, take the highest power of each prime factor:
- For 2, the maximum power is $2^1$ (from both 14 and 42).
- For 3, the maximum power is $3^1$ (from 42).
- For 7, the maximum power is $7^1$ (from both).
- Calculate the LCM Now, multiply all these together: $$ LCM = 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 $$ Calculating this: $$ 2 \times 3 = 6 \ 6 \times 7 = 42 $$
The least common multiple (LCM) of 14 and 42 is 42.
The least common multiple of 14 and 42 is ( 42 ).
More Information
The least common multiple (LCM) is useful in many applications, such as finding common denominators in fractions or solving problems that involve repeat cycles. The LCM of two numbers is the smallest number that is a multiple of both.
Tips
- Forgetting to include all prime factors or their highest powers when calculating the LCM.
- Confusing LCM with the greatest common divisor (GCD), which involves finding the largest number that divides both without a remainder.
