Finding the Least Squared Number Exactly Divisible by 8, 12, 15, and 20

In the world of number theory, finding the least square number that is exactly divisible by a set of numbers is a fascinating problem. This article will explore how to find such a number, specifically for the numbers 8, 12, 15, and 20.

Understanding the Problem

The goal is to determine the least number that is a perfect square and is divisible by each of the given numbers: 8, 12, 15, and 20. A perfect square, as its name suggests, is a number that is the square of an integer. This means that all the prime factors in its prime factorization must have even exponents.

Prime Factorization and Least Common Multiple (LCM)

Let's start by examining the prime factorization of each number:

t8 2^3 t12 2^2 × 3 t15 3 × 5 t20 2^2 × 5

The least common multiple (LCM) of these numbers is the smallest number that is a multiple of all of them. The LCM is calculated by taking the highest power of each prime that appears in the factorizations:

LCM 2^3 × 3 × 5 120

Determining the Least Squared Number

Note that the LCM, 120, alone is not a perfect square because the exponents of the primes in its prime factorization are not all even. To make it a perfect square, we need to adjust the exponents of the primes so that they are all even. Specifically, we need to multiply the LCM by 2, 3, and 5 to get all the exponents to be even:

2^3 × 3 × 5 2 × 2^2 × 3 × 5 30

So, we calculate:

120 × 30 3600

Therefore, the smallest square number divisible by 8, 12, 15, and 20 is 3600. To confirm, we can verify that:

t3600 is a perfect square: tsqrt(3600) 60

And 3600 is divisible by:

t3600 ÷ 8 450 t3600 ÷ 12 300 t3600 ÷ 15 240 t3600 ÷ 20 180

Using the J Programming Language

The J programming language provides a succinct way to solve this problem. Here is a step-by-step example:

??, ( 8, 12, 15, 20 )? LCM? ?????: t./a~./08 12 15 20/a.:1 to 50 t144 t

?? ??: 144? ??? 8, 12, 15,? 20? ?? ? ?? ??????.

Conclusion

By following the steps outlined in this article, you can determine the least square number exactly divisible by any set of numbers. The method involves finding the LCM and then ensuring that all the prime factors have even exponents. In the case of 8, 12, 15, and 20, the answer is 3600. This problem has applications in cryptography, number theory, and computer science, making it an important topic to understand.