How Many 5-Digit Numbers Are Divisible by 4, 5, 6, 7, 8, 9, 10, and 11?

Understanding the least common multiple (LCM) and its application in number theory can often seem like a challenge. However, it is a powerful tool for solving problems involving divisibility. One intriguing question is how many 5-digit numbers are divisible by 4, 5, 6, 7, 8, 9, 10, and 11. This article will explore this problem and provide a detailed solution.

The Least Common Multiple (LCM)

The least common multiple (LCM) of a set of numbers is the smallest positive integer that is divisible by each of the numbers in the set. In this problem, we are given the numbers 4, 5, 6, 7, 8, 9, 10, and 11. To find the LCM, we can break down each number into its prime factors:

4 22 5 5 6 2 × 3 7 7 8 23 9 32 10 2 × 5 11 11

The LCM is found by taking the highest powers of all prime factors present in the set. Therefore, the LCM of these numbers is:

5 × 7 × 8 × 9 × 11 27720

This product itself is a 5-digit number, which helps narrow down the possibilities.

Divisibility by 4, 6, and 10

Since 4, 6, and 10 are factors of other numbers in the set (4 is a factor of 8, 6 is a factor of both 6 and 9, and 10 is a factor of 10), we can ignore them for the LCM calculation. However, this does not affect the divisibility aspect. Specifically:

27720 is divisible by 4 because it contains the prime factor 23. 27720 is divisible by 6 because it contains the prime factors 2 and 3. 27720 is divisible by 10 because it contains the prime factors 2 and 5.

Thus, 27720 is divisible by 4, 6, and 10.

Checking for Other 5-Digit Numbers

Next, we need to check if there are other 5-digit numbers that are divisible by 4, 5, 6, 7, 8, 9, and 11. We start by multiplying 27720 by these numbers:

27720 × 4 110880, which is a 6-digit number and thus not a 5-digit number. 27720 × 2 55440, which is a 5-digit number. 27720 × 3 83160, which is also a 5-digit number.

Since 27720 itself is a 5-digit number, and we have found two other 5-digit numbers (55440 and 83160), these are the only 5-digit numbers that are divisible by all given numbers.

Conclusion

After checking all possibilities, the answer to the question is clear: there are exactly three 5-digit numbers divisible by 4, 5, 6, 7, 8, 9, 10, and 11, which are 27720, 55440, and 83160. This demonstrates the power of the least common multiple in solving problems involving divisibility.