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least common multiple of 21 and 49

Understand the Problem

The question is asking how to find the least common multiple (LCM) of the two numbers 21 and 49. This involves identifying the smallest multiple that both numbers share.

Answer

The least common multiple of 21 and 49 is $147$.
Answer for screen readers

The least common multiple (LCM) of 21 and 49 is 147.

Steps to Solve

  1. List the multiples of each number

First, we will find the multiples of both numbers.

For 21, the first few multiples are: $$ 21, 42, 63, 84, 105, ... $$

For 49, the first few multiples are: $$ 49, 98, 147, 196, 245, ... $$

  1. Identify the common multiples

Next, we look for multiples that appear in both lists.

The multiples we found are:

  • 21: $21, 42, 63, 84, 105, ...$
  • 49: $49, 98, 147, 196, 245, ...$

The smallest multiple that is common to both lists is not immediately visible, so we will need to use another method.

  1. Use the prime factorization method

Now, let's find the prime factorization of both numbers.

For 21: $$ 21 = 3 \times 7 $$

For 49: $$ 49 = 7^2 $$

  1. Calculate the LCM using the highest power of each prime factor

To find the LCM, we take the highest power of each prime factor:

  • From 21: $3^1$ and $7^1$
  • From 49: $7^2$

Now, we calculate the LCM: $$ LCM = 3^1 \times 7^2 = 3 \times 49 = 147 $$

Thus, the least common multiple of 21 and 49 is 147.

The least common multiple (LCM) of 21 and 49 is 147.

More Information

The least common multiple (LCM) is useful in various mathematical contexts, such as finding common denominators in fractions or solving problems involving periodic events.

Tips

  • Forgetting to include all prime factors when calculating the LCM.
  • Confusing the LCM with the greatest common divisor (GCD), which is the largest factor shared between numbers, rather than the smallest multiple.
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