lcm of 35 and 50

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 35 and 50. This means we need to find the smallest positive integer that is divisible by both 35 and 50.

Answer

$350$

Steps to Solve

Find the prime factorization of each number

Start by breaking down each number into its prime factors.

The prime factorization of 35 is: $$ 35 = 5 \times 7 $$
The prime factorization of 50 is: $$ 50 = 2 \times 5^2 $$

Identify the highest power of each prime factor

Now, we take each prime number found in the factorizations and determine the highest power of that prime that appears in either number.

For the prime factor 2: highest power is $2^1$ from 50.
For the prime factor 5: highest power is $5^2$ from 50.
For the prime factor 7: highest power is $7^1$ from 35.

Calculate the LCM

Now we multiply these highest powers together to find the LCM: $$ LCM = 2^1 \times 5^2 \times 7^1 $$ Calculating this: $$ LCM = 2 \times 25 \times 7 $$

Perform the multiplication

Carrying out the multiplication step-by-step:

First, calculate $2 \times 25 = 50$
Then, calculate $50 \times 7 = 350$

Thus, the least common multiple of 35 and 50 is 350.

The least common multiple (LCM) of 35 and 50 is $350$.

More Information

The least common multiple is particularly important in solving problems related to fractions, such as adding or subtracting fractions with different denominators, as it helps find a common denominator. LCM is also used in scheduling problems where events occur at different intervals.

Tips

Confusing LCM with GCD (greatest common divisor). Remember, LCM is the smallest multiple, while GCD is the largest divisor.
Forgetting to use the highest power of each prime factor, which can lead to an incorrect answer.

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