Finding the Smallest Five-Digit Number Divisible by Given Numbers

The goal is to identify the smallest integer that has five digits and is exactly divisible by 12, 16, 24, and 28.

Step 1: Calculate the Least Common Multiple (LCM)

To find a number divisible by all the given numbers (12, 16, 24, 28), we first calculate their LCM. This is the smallest positive number that is a multiple of all of them.

Find the prime factorization of each number:

• $12 = 2^2 \times 3$
• $16 = 2^4$
• $24 = 2^3 \times 3$
• $28 = 2^2 \times 7$

The LCM is the product of the highest powers of all prime factors involved:

LCM$(12, 16, 24, 28) = 2^4 \times 3^1 \times 7^1 = 16 \times 3 \times 7 = 336$.

Step 2: Find the Smallest Five-Digit Multiple of the LCM

The smallest five-digit number is 10000.

To find the smallest five-digit number divisible by 336, we divide 10000 by 336 and find the next whole number multiple.

$ \frac{10000}{336} \approx 29.76 $

The next integer after 29.76 is 30. This means the 30th multiple of 336 is the smallest multiple that is a five-digit number.

Calculate the 30th multiple:

$ 30 \times 336 = 10080 $

Result

The smallest five-digit number divisible by 12, 16, 24, and 28 is 10080.