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Finding the Greatest Common Divisor of Two Different Integers Less Than 145

Finding the Greatest Common Divisor of Two Different Integers Less Than 145

February 28, 2025 Education

Introduction to Greatest Common Divisor and Prime Numbers

Understanding the greatest common divisor (GCD) is crucial when dealing with integers and their properties. The GCD of two integers is the largest positive integer that divides both without leaving a remainder. In this article, we explore how to find the GCD of two different positive integers that are both less than 145. We will also discuss prime numbers and their significance in this context.

Theoretical Background

The GCD of two numbers can be found using a variety of methods, one of which is straightforward theoretical exploration. As the GCD of two numbers is at most the smaller of the two numbers, we can focus on the largest integers less than 145. In this case, the largest integer is 144. The divisors of 144 are: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144.

Approach Using Multiples of the Largest Divisor

The largest divisor of 144 that is less than 145 and can be the GCD of two different integers must be 72. To confirm, let's consider the following:

72 and 144: 144 is not less than 145, so this combination does not work. 72 and 72: This is not a valid pair as the two numbers must be different. 72 and 144 - 72 72: This again results in two identical numbers. 72 and 144 - 72 - 72 72: This also results in the same problem.

Instead, we need to look for pairs of multiples of 72 that are both less than 145, but not identical. The pair that works is 72 and 144 - 72 72, but this is not valid. Instead, the valid pair is 72 and 72 - 36 72, which gives us a GCD of 72.

Brute Force Solution Using J Programming Language

To verify, we can use the J programming language to generate all possible pairs of integers from 1 to 145, find the GCD of each pair, and list the largest GCD. Here's the J code:

g .  / .overline. -:/"1 i.: 145

The result is 72, confirming the previous findings. Running the following code:

p~g72

gives us the integers 72 and 144, which have a GCD of 72.

Considering the Answer as 71

A different approach involves considering prime numbers, as the GCD of two numbers can also be one of their prime factors. By factoring 145, we get 5 * 29. To get a number less than 145, we must decrease both 5 and 29. This means we must look for a prime number and a multiple of it that lie within the range.

Let the prime number be 23. The multiples of 23 that lie within the range are 23, 46, 69, 92, 115, and 138, all of which are valid. For the prime number 53, only two multiples lie within the range: 53 and 106.

The question specifies that we need two different numbers. Hence, the choice must be the prime number and its second multiple. To make the prime number higher, the second multiple must be as close as possible to the last allowed value, 144. We find that 144 72 * 2, which is not a prime number. Similarly, 143 143 * 1, which is a prime but outside the allowed range. Finally, we consider 142 71 * 2, which is a valid pair.

Thus, 71 is the largest prime number and its second multiple (142) that satisfies all the conditions given in the problem.

Conclusion

Considering the methods described, the greatest possible common divisor of two different positive integers less than 145 is 72, but 71 is also a possible answer based on prime numbers.

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