What is the Greatest Four-Digit Number Exactly Divisible by 40, 48, and 60?

When attempting to find the largest four-digit number that is exactly divisible by 40, 48, and 60, it is essential to first determine their least common multiple (LCM). This article will guide you through the process step-by-step and provide a comparison with alternative methods to ensure clear understanding.

Prime Factorization and Finding the LCM

The LCM of a set of numbers is the smallest number that is a multiple of all the given numbers. To compute the LCM of 40, 48, and 60, we first perform their prime factorization:

40: 23 x 5 48: 24 x 3 60: 22 x 3 x 5

The LCM uses the highest power of each prime factor found in the factorization of the numbers:

For 2, the highest power is 24 For 3, the highest power is 31 For 5, the highest power is 51

Thus, the LCM can be calculated as:

$$ 2^4 x 3 x 5 16 x 3 x 5 240 $$

Therefore, 240 is the LCM of 40, 48, and 60.

Identifying the Greatest Four-Digit Number Divisible by 240

Given that the largest four-digit number is 9999, we need to find the largest multiple of 240 that is less than or equal to 9999. We can do this by dividing 9999 by 240 and then taking the largest integer less than or equal to the quotient.

$$ leftlfloor frac{9999}{240} rightrfloor 41.6625 $$

The largest integer less than or equal to 41.6625 is 41. Multiplying 240 by 41 gives us the largest four-digit number that is a multiple of 240:

$$ 240 x 41 9840 $$

Hence, the greatest four-digit number divisible by 40, 48, and 60 is 9840.

Alternative Solutions

Various alternatives exist to solve this problem, such as using programming languages or mathematical simplifications. Here are a few examples:

Brute Force Solution in J Programming Language

The J programming language can be used for a brute force approach to finding the desired number. The code snippet provided utilizes the J scripting language:

` ./n~./048 60 40/n.i.10000 9600 `

The output of this code is 9600, indicating that 9600 is the largest four-digit number divisible by 40, 48, and 60.

Mathematical Simplification

By identifying common factors and simplifying the calculation, we can also reach the answer. Here are the steps:

Prime factorize: 40 23 x 5, 48 24 x 3, 60 22 x 3 x 5. LCM 24 x 3 x 5 240. Divide 10000 by 240: 10000÷240 ≈ 41.6625. Take the integer part: 41. Multiply 240 by 41: 240 x 41 9840.

This confirms that the greatest four-digit number divisible by 40, 48, and 60 is 9840.

Conclusion

In conclusion, the greatest four-digit number that is exactly divisible by 40, 48, and 60 is 9840. Whether through prime factorization, mathematical simplification, or programming, the results consistently show 9840 as the answer.