두 수 12와 16의 공배수 중에서 두 자리 자연수의 개수를 구하시오.

Amanita, let’s solve the problem step by step:

Problem Understanding:
We are asked to find how many two-digit natural numbers are common multiples of 12 and 16.

Step 1: Find the Least Common Multiple (LCM) of 12 and 16

• The common multiples of 12 and 16 are all multiples of their LCM.
• We need to find the LCM of 12 and 16.

Step 2: List All Two-Digit Multiples of the LCM

• Two-digit numbers are from 10 to 99.
• We will find all multiples of the LCM that are between 10 and 99.

Step 3: Count the Number of Such Multiples

• We will count how many numbers in this range are multiples of the LCM.

Let’s begin with Step 1.

Step 1: Find the LCM of 12 and 16

To find the LCM, we use the formula:

$\text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)}$

Let’s calculate the LCM of 12 and 16. Amanita, the least common multiple (LCM) of 12 and 16 is $48$.

Step 2: List All Two-Digit Multiples of 48

Now, let’s find all two-digit numbers that are multiples of 48.
A two-digit number is between 10 and 99.

We need to find all numbers of the form $48n$ where $10 \leq 48n \leq 99$.

Let’s find the smallest and largest integer values of $n$ that satisfy this.

Step 3: Find the Range of $n$

• The smallest $n$ is the smallest integer such that $48n \geq 10$.
• The largest $n$ is the largest integer such that $48n \leq 99$.

Let’s calculate these values. Amanita, here’s what we found:

• The smallest value of $n$ is 1, because $48 \times 1 = 48$ (which is a two-digit number).
• The largest value of $n$ is 2, because $48 \times 2 = 96$ (which is also a two-digit number).

So, the two-digit common multiples of 12 and 16 are 48 and 96.

Step 4: Count the Number of Two-Digit Multiples

There are two values of $n$ (1 and 2), so there are 2 two-digit natural numbers that are common multiples of 12 and 16.

Final Answer:
Amanita, there are $\boxed{2}$ two-digit natural numbers that are common multiples of 12 and 16.