For how many two-digit positive integers does the product of the digits come out to $12$?

Let's determine how many two-digit positive integers have a product of digits equal to $12$.

Let the two-digit number be represented as $ \overline{ab} $, where $ a $ is the tens digit and $ b $ is the units digit. We need to find all pairs $(a, b)$ such that:

\[
a \times b = 12
\]

The possible pairs of digits $(a, b)$ that multiply to 12 are:

\[
(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1)
\]

However, $a$ and $b$ must each be a single digit (i.e., between 1 and 9 inclusive), so we discard any pairs where $a$ or $b$ is greater than 9. Therefore, the valid pairs are:

\[
(2, 6), (3, 4), (4, 3), (6, 2)
\]

Each of these pairs corresponds to a different two-digit number:

\[
26, 34, 43, 62
\]

So, there are $\boxed{4}$ two-digit positive integers where the product of the digits is 12.

Would you like further details or have any other questions? Here are some related problems to explore:

1. How many two-digit numbers have the product of the digits equal to 16?
2. How many two-digit numbers have the sum of the digits equal to 12?
3. What is the largest two-digit number whose digits have a product of 12?
4. Can any three-digit numbers have a product of the digits equal to 12?
5. How many two-digit numbers have the product of the digits equal to 18?

**Tip:** When solving problems involving digits, ensure the conditions fit within the range of single-digit numbers (0-9).

Discover how many two-digit positive integers have a product of the digits equal to 12 with a step-by-step solution. This problem explores basic multiplication concepts and is suitable for students in Grades 4-6.

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