What is the Least Square Number Divisible by 35, 15, and 49?

In this article, we will explore the process of finding the smallest square number that is divisible by 35, 15, and 49. We will start by breaking down each number into its prime factors, determining the least common multiple (LCM), and then finding the smallest perfect square that is a multiple of that LCM.

Step-by-Step Solution

Step 1: Factor Each Number

The first step is to factor each of the numbers:

35 5 × 7 15 3 × 5 49 72

Step 2: Find the Least Common Multiple (LCM)

To find the LCM, we take the highest power of each prime factor that appears in the factorizations:

Highest power of 3: 31 Highest power of 5: 51 Highest power of 7: 72

Therefore, the LCM is calculated as:

LCM 31 × 51 × 72 3 × 5 × 49 735

Step 3: Find the Smallest Perfect Square Multiple of 735

The next step is to ensure that all prime factors have even exponents to form a perfect square. The prime factorization of 735 is:

735 31 × 51 × 72

We need to increase the exponent of 3 from 1 to 2, making it 32. We need to increase the exponent of 5 from 1 to 2, making it 52. The exponent of 7 is already 2, which is even.

Therefore, we multiply 735 by 3 and 5:

Perfect square 735 × 3 × 5 735 × 15 11025

The least square number divisible by 35, 15, and 49 is: 11025.

Additional Results

There is also a way to solve this problem using the J programming language:

 {.a~./035 15 49/a:.:1i.100011025

This algorithm yields 11025 as the smallest square number divisible by 35, 15, and 49. Other related results include 44100, 99225, 176400, and so on.

Civilization and Mental Processes

In his insightful quote, Alfred North Whitehead articulates the importance of freeing our minds to perform complex operations without conscious thought:

Civilization advances by extending the number of important operations which we can perform without thinking of them. Alfred North Whitehead

This concept is particularly relevant when dealing with mathematical operations, where understanding and applying the correct methods can lead to efficient problem-solving, such as finding the least square number divisible by multiple integers.