Introduction

Identifying the least two-digit number that has 2, 5, and 8 as factors is a straightforward yet intriguing problem in number theory. This guide will step through the process of finding this number, explore the mathematical reasoning behind it, and provide a practical example using the J programming language.

Understanding the Problem

The primary goal is to find the smallest two-digit number (between 10 and 99) that can be evenly divided by 2, 5, and 8 without leaving any remainder. To do so, we need to understand the concept of the least common multiple (LCM), which is the smallest number that is a multiple of the given numbers.

Prime Factorization and LCM Calculation

First, let's break down the numbers into their prime factors:

2 is already a prime number. 5 is already a prime number. 8 can be expressed as 23.

The LCM is found by taking the highest power of each prime factor present in the numbers. For 2, the highest power is 23; for 5, it is 51.

LCM 23 × 51 8 × 5 40

Verification and Brute Force Solution

To verify, we can use the J programming language to find the least number that satisfies the condition:

{.n~./02 5 8/n.10 to 90

The result is 40.

Without using code, we can check manually:

10 is not a multiple of 8. 20 is not a multiple of 8. 30 is not a multiple of 8. 40 is a multiple of 2, 5, and 8.

Thus, the least two-digit number that has 2, 5, and 8 as factors is 40.

Further Exploration: Least Two-Digit Number with Different Factors

Let's consider another set of factors: 2, 5, 7, and 9. To find the least two-digit number that these factors share as common multiples, we need to calculate the LCM of these numbers.

Prime Factorization of the New Factors

2 is already a prime number. 5 is already a prime number. 7 is already a prime number. 9 can be expressed as 32.

The LCM is found by taking the highest power of each prime factor present in the numbers. For 3, the highest power is 32.

LCM 2 × 5 × 7 × 9 630

630 is a valid number, but it is a three-digit number. The smallest two-digit number that is a multiple of 2, 5, 7, and 9 is the smallest multiple of 630 that is a two-digit number, which is 90 (630 ÷ 7 90).

Verification of the Solution

90 ÷ 2 45 (no remainder) 90 ÷ 5 18 (no remainder) 90 ÷ 7 12.857 (not an integer, but 90 ÷ 9 10, which is an integer)

Therefore, the answer is the least two-digit number that has 2, 5, 7, and 9 as factors, which is 90.

Conclusion

In this guide, we've explored how to find the least two-digit number that has specific factors, using the concept of least common multiples (LCM) and prime factorization. We've demonstrated both the mathematical reasoning and the practical application of these concepts in a clear and concise manner. Whether you're working on a math problem or a coding challenge, understanding these principles can be incredibly valuable.

Additional Resources

For further study, consider exploring more advanced topics in number theory or checking out online resources like Khan Academy and Wolfram Alpha for additional practice and explanations.