We are tasked with finding the smallest two-digit natural number $N$ such that:

1. $N$ is divisible by 3.
2. Reversing the digits of $N$ produces a new number divisible by 2.

Step 1: Represent the two-digit number $N$

Let $N$ be represented as $10a + b$, where $a$ is the tens digit and $b$ is the ones digit. The reversed number would be $10b + a$.

Step 2: Condition 1: $N$ is divisible by 3

For $N$ to be divisible by 3, the sum of its digits, $a + b$, must be divisible by 3. Therefore, we require: $a + b \equiv 0 \pmod{3}$

Step 3: Condition 2: The reversed number is divisible by 2

For the reversed number $10b + a$ to be divisible by 2, the ones digit, which is $a$, must be even. Therefore, $a$ must be an even number. This gives us possible values of $a = 0, 2, 4, 6, 8$.

Step 4: Find the smallest $N$ satisfying both conditions

We now check for the smallest possible value of $N = 10a + b$ such that:

• $a + b \equiv 0 \pmod{3}$
• $a$ is even.

Case 1: $a = 0$

If $a = 0$, then $b$ must satisfy $0 + b \equiv 0 \pmod{3}$, which implies $b \equiv 0 \pmod{3}$. The smallest $N$ with $a = 0$ and $b \equiv 0 \pmod{3}$ is $N = 30$. However, this is not a two-digit number, so we discard it.

Case 2: $a = 2$

If $a = 2$, then $2 + b \equiv 0 \pmod{3}$, which implies $b \equiv 1 \pmod{3}$. Therefore, $b = 1, 4, 7$.

• For $b = 1$, $N = 21$, and the reversed number is $12$, which is divisible by 2. So, $N = 21$ satisfies all conditions.

Since 21 is a two-digit number, divisible by 3, and its reversed number 12 is divisible by 2, the smallest possible value of $N$ is: $\boxed{21}$

Would you like to go over any part of this solution or have further questions?

5 Related Questions:

1. Can a number divisible by both 2 and 3 be divisible by 6?
2. What is the general rule for divisibility by 2 and 3?
3. How do the digits of a number relate to its divisibility by 9?
4. How can you quickly find the reverse of any two-digit number?
5. What would change in the problem if $N$ had three digits?

Tip:

When checking divisibility rules, breaking down numbers into their digits often reveals useful patterns.