Greatest Five Digit Nummber Divislbe By 72

Finding the Greatest Five-Digit Number Divisible by 72

Finding the largest five-digit number divisible by 72 might seem like a simple math problem, but understanding the process can be surprisingly insightful, especially when considering broader applications in computer science and number theory. This article will guide you through the steps, explaining the logic behind the solution and offering valuable insights into divisibility rules. This is perfect for anyone interested in number theory, programming, or simply sharpening their mathematical skills.

The largest five-digit number is 99,999. Our goal is to find the largest number less than or equal to 99,999 that is perfectly divisible by 72. This means the remainder when dividing by 72 must be zero.

Understanding Divisibility by 72

Before we dive into the calculation, let's refresh our understanding of divisibility rules. A number is divisible by 72 if it's divisible by both 8 and 9, since 72 = 8 x 9. This significantly simplifies our task.

Divisibility by 8: A number is divisible by 8 if the last three digits are divisible by 8.
Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.

The Calculation

We can use these rules to efficiently find our answer. Instead of brute-forcing the division of every number down from 99,999, we'll use a more strategic approach.

Start with the largest five-digit number: 99,999
Check for divisibility by 8: The last three digits are 999. 999 divided by 8 leaves a remainder. Therefore, 99,999 is not divisible by 8, and thus not divisible by 72.
Iterate downwards: We need to find a number slightly smaller than 99,999 that meets our divisibility criteria. We can do this by repeatedly subtracting 72 until we find a number with a remainder of zero. However, a more efficient method is to use integer division.
Integer Division: Divide 99,999 by 72 using integer division (which discards the remainder). This gives us the quotient. Then, multiply this quotient by 72. This will give us the largest multiple of 72 that is less than or equal to 99,999.

99999 // 72 = 1388 (integer division) 1388 * 72 = 99936

Therefore, the greatest five-digit number divisible by 72 is 99,936.

Verification

Let's verify our result using our divisibility rules:

Divisibility by 8: The last three digits are 936. 936 / 8 = 117. Therefore, 99,936 is divisible by 8.
Divisibility by 9: The sum of the digits is 9 + 9 + 9 + 3 + 6 = 36. 36 / 9 = 4. Therefore, 99,936 is divisible by 9.

Since 99,936 is divisible by both 8 and 9, it is divisible by 72.

Conclusion

By leveraging the divisibility rules for 8 and 9, we efficiently determined that 99,936 is the greatest five-digit number divisible by 72. This approach showcases the power of understanding fundamental mathematical principles to solve seemingly complex problems. This method is easily adaptable to finding the largest n-digit number divisible by any composite number.