What is the Number Divisible by All Digits?

The smallest positive integer divisible by all single-digit numbers (1 through 9) is 2520.

Explanation:

This question refers to finding the Least Common Multiple (LCM) of the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9. The LCM is the smallest number that is a multiple of each of these numbers.

To calculate the LCM, we can use prime factorization:

• 1 = 1
• 2 = 2
• 3 = 3
• 4 = 2²
• 5 = 5
• 6 = 2 * 3
• 7 = 7
• 8 = 2³
• 9 = 3²

The LCM is found by taking the highest power of each prime factor present:

• 2³ = 8
• 3² = 9
• 5 = 5
• 7 = 7

Therefore, the LCM is 2³ 3² 5 7 = 8 9 5 7 = 2520.

Significance of 2520

As the reference link suggests, 2520 is:

• Half of 7! (7 factorial, or 7 6 5 4 3 2 1 = 5040)
• A superior highly composite number.
• A colossally abundant number.

These properties indicate that 2520 has a large number of divisors compared to numbers close to it, making it divisible by many different integers.