Understanding the Problem: Finding the Smallest Number Divisible by 18, 36, 32, and 27 When Decreased by 3

In this article, we will explore the process of finding the smallest number, x, which, when decreased by 3, is completely divisible by 18, 36, 32, and 27. This involves a series of steps, including prime factorization and calculating the least common multiple (LCM).

Step-by-Step Breakdown

Step 1: Prime Factorization

To find the LCM, we first need to determine the prime factorizations of each given number:

18: 18 2 × 3^2 36: 36 2^2 × 3^2 32: 32 2^5 27: 27 3^3

Step 2: Determine the LCM

The LCM is found by taking the highest power of each prime factor that appears in the factorizations. This means:

The highest power of 2 is 2^5 from 32. The highest power of 3 is 3^3 from 27.

The LCM is then calculated as follows:

LCM 2^5 × 3^3

Step 3: Calculate the LCM

Calculating the values:

2^5: 2^5 32 3^3: 3^3 27

Therefore, the LCM is:

LCM 32 × 27 864

Step 4: Find the Smallest x

Since we need x - 3 864k for some integer k, the smallest positive x occurs when k 1:

x - 3 864 x 864 3 867

Conclusion

Therefore, the smallest number which when decreased by 3 is completely divisible by 18, 36, 32, and 27 is:

boxed{867}

Revisiting the Steps to Confirm

First, we find the LCM of 18, 36, 32, and 27:

18 2×3×3 36 2×2×3×3 32 2×2×2×2×2 27 3×3×3

LCM 864

864 is the smallest integer which is divisible by 18, 36, 32, and 27.

Hence, the smallest integer which needs to be decreased by 3 is 864 - 3 867.

867 when decreased by 3 will be divisible by 18, 36, 32, and 27.

Factors

Factors of:

18 2×3×3 36 2×2×3×3 32 2×2×2×2×2 27 3×3×3

LCM 2×2×2×2×2×3×3×3 864

864 - 3 867

The number you are seeking is 867.

Additional Information

Starting from the beginning, we calculate the least common multiple (LCM) of the given numbers as follows:

18 2×3×3 36 2×2×3×3 32 2×2×2×2×2 27 3×3×3