Introduction

This guide will help you understand the process of identifying numbers between 1 and 199 that are divisible by both 5 and 7. We'll use mathematical concepts such as Least Common Multiple (LCM) and prime numbers to solve this problem efficiently.

Understanding the Problem

The problem asks us to find how many numbers between 1 and 199 are divisible by both 5 and 7. A number divisible by both 5 and 7 must be a multiple of their least common multiple (LCM).

Mathematical Explanation

To solve this problem, we first need to find the LCM of 5 and 7. Since 5 and 7 are prime numbers, their LCM is simply their product:

Step 1: Calculate the LCM of 5 and 7

The LCM of two numbers (a) and (b) is given by:

LCM((a), (b)) (frac{a times b}{text{GCD}(a, b)})

Since 5 and 7 are prime and have no common factors other than 1, the GCD(5, 7) is 1. Therefore:

LCM(5, 7) (frac{5 times 7}{1} 35)

A number divisible by both 5 and 7 must, therefore, be a multiple of 35.

Step 2: Determine the Multiples of 35 Between 1 and 199

We need to find how many multiples of 35 lie between 1 and 199. This can be done by dividing 199 by 35 and taking the floor of the result. The formula to find the number of multiples of 35 between 1 and 199 is:

( leftlfloor frac{199}{35} rightrfloor )

Performing the division:

( frac{199}{35} approx 5.6857 )

Taking the floor of 5.6857 gives us:

floor(5.6857) 5

So, there are 5 multiples of 35 between 1 and 199.

Step 3: List the Multiples of 35

The multiples of 35 between 1 and 199 are:

35 (times) 1 35 35 (times) 2 70 35 (times) 3 105 35 (times) 4 140 35 (times) 5 175

Brute Force Solution Using J Programming Language

Using the J programming language, we can find the solution efficiently:

n ~./ 0  5 7 / n . 1 to 19935 70 105 140 175

The result confirms that there are 5 numbers between 1 and 199 that are divisible by both 5 and 7.

Conclusion

In conclusion, the numbers between 1 and 199 that are divisible by both 5 and 7 are 35, 70, 105, 140, and 175. Therefore, the total number of such numbers is 5.

Keywords

LCM, Divisibility, Prime Numbers