Finding the Largest Four-Digit Number Divisible by 12, 15, and 18

The objective is to find the largest possible number that has exactly four digits and can be evenly divided by 12, 15, and 18.

Step 1: Calculate the Least Common Multiple (LCM)

First, find the LCM of the divisors 12, 15, and 18. This LCM will be the smallest number divisible by all three.

• Prime factorization of 12 is $2^2 \times 3$.
• Prime factorization of 15 is $3 \times 5$.
• Prime factorization of 18 is $2 \times 3^2$.

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

LCM$(12, 15, 18) = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180$.

This means any number divisible by 12, 15, and 18 must be a multiple of 180.

Step 2: Determine the Largest Four-Digit Number

The range of four-digit numbers is from 1000 to 9999. The largest number in this range is 9999.

Step 3: Find the Largest Multiple of the LCM within Four Digits

To find the largest four-digit number divisible by 180, divide the largest four-digit number (9999) by the LCM (180).

$ \frac{9999}{180} \approx 55.55 $

The largest integer multiplier less than or equal to 55.55 is 55.

Multiply this integer by the LCM to get the largest four-digit multiple:

$ 55 \times 180 = 9900 $

This result, 9900, is the largest four-digit number divisible by 180, and therefore divisible by 12, 15, and 18.