What would be the L.C.M. of 8, 12, 18?

To find the least common multiple (L.C.M.) of the numbers 8, 12, and 18, we can use a systematic approach. The L.C.M. is the smallest number that is a multiple of each of the given numbers. There are a couple of methods to calculate this, but one effective way is to use the prime factorization of each number.

Step-by-Step Prime Factorization

Let's break down each number into its prime factors:

• 8: The prime factorization of 8 is 2 × 2 × 2, or 2³.
• 12: The prime factorization of 12 is 2 × 2 × 3, or 2² × 3¹.
• 18: The prime factorization of 18 is 2 × 3 × 3, or 2¹ × 3².

Identifying the Highest Powers

Next, we need to identify the highest powers of each prime factor that appear in these factorizations:

• For the prime number 2, the highest power is 2³ (from 8).
• For the prime number 3, the highest power is 3² (from 18).

Calculating the L.C.M.

Now, we can calculate the L.C.M. by multiplying these highest powers together:

L.C.M. = 2³ × 3²

Calculating this gives:

• 2³ = 8
• 3² = 9

Now, multiply these results:

L.C.M. = 8 × 9 = 72

Final Result

Thus, the least common multiple of 8, 12, and 18 is 72.

Verification

To ensure that 72 is indeed a multiple of each of the original numbers, we can check:

• 72 ÷ 8 = 9 (which is an integer)
• 72 ÷ 12 = 6 (which is an integer)
• 72 ÷ 18 = 4 (which is an integer)

Since 72 is divisible by all three numbers, we have confirmed that our calculation is correct. The L.C.M. of 8, 12, and 18 is indeed 72.