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To find the Least Common Multiple (LCM) of two numbers, the first crucial step is to perform prime factorization. Prime factorization means breaking down a number into its basic building blocks, which are prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself.

Consider the numbers 6 and 21. For 6, we find that it can be divided by 2 (which is a prime number) to give us 3, another prime number. So, 6 can be expressed as the product of 2 and 3, or 6 = 2 × 3.

For 21, we can divide it by 3, a prime number, to give us 7, another prime number. Therefore, 21 is 21 = 3 × 7.

By breaking down both numbers into these prime factors, we're preparing to compare them in the next steps.

Once we have the prime factors of each number, the next step is to identify the highest powers of each unique prime factor. This involves listing all the prime factors and selecting the highest powers present in the factorizations.

For our numbers, 6 and 21, we found the prime factors to be 2, 3, and 7. The number 6 breaks down to 2 × 3, while 21 breaks down to 3 × 7.

We need to consider the highest power of each of these prime factors.

• 2: Only appears in 6, hence the power is 2¹.
• 3: Appears in both 6 and 21, hence we take 3¹ as the highest power.
• 7: Only appears in 21, hence the power is 7¹.

Recognizing the highest power of each prime factor ensures that the calculated LCM will evenly divide both original numbers.

The final step in finding the LCM is to multiply the highest powers of all the prime factors together.

For the numbers 6 and 21, we identified the following highest powers:

• 2¹
• 3¹
• 7¹

This means the LCM can be calculated as:


Therefore:


which gives us:

Finally:

Therefore, the LCM of 6 and 21 is 42.

Multiplying these factors ensures that both 6 and 21 can divide the LCM without any remainder, making 42 the smallest number that satisfies this condition.