The problem requires using the numbers 3, 4, and 5 exactly once in the following calculations to make the result an even number: $5 \times \square$ $12 + \square$ $9 + \square$

1. Problem Description

The problem requires using the numbers 3, 4, and 5 exactly once in the following calculations to make the result an even number:

$5 \times \square$

$12 + \square$

$9 + \square$

2. Solution Steps

We need to fill in the blanks with the numbers 3, 4, and 5 such that the results of the calculations are even.

* Consider $5 \times \square$.

* If we put 3 in the blank, $5 \times 3 = 15$, which is odd.

* If we put 4 in the blank, $5 \times 4 = 20$, which is even.

* If we put 5 in the blank, $5 \times 5 = 25$, which is odd.

Therefore, we must have $5 \times 4 = 20$.

* Now we have the numbers 3 and 5 left. Let's consider $12 + \square$.

* If we put 3 in the blank, $12 + 3 = 15$, which is odd.

* If we put 5 in the blank, $12 + 5 = 17$, which is odd.

Neither of these work, so let's try a different approach.

Let's look at the sums:

$12 + \square$

$9 + \square$

For the first sum to be even, we need to add an even number to

1

2. For the second sum to be even, we need to add an odd number to

9. Therefore, we need to place 3 and 5 in the sums and 4 in the product.

So, $5 \times 4 = 20$

Then, we need to determine where 3 and 5 should go:

$12 + \square$

$9 + \square$

Since we need to add an odd number to 9, we must have $9 + 3 = 12$.

Then we must have $12 + 5 = 17$. But the answer needs to be an even number.

Let's consider a different approach.

$5 \times 4 = 20$, which is even. So 4 is the number in the first blank.

For the other two calculations, the answers must be even:

$12 + \square = \text{even}$

$9 + \square = \text{even}$

Since 12 is even, we need to add an even number to it to get an even number. Since 9 is odd, we need to add an odd number to it to get an even number.

We are left with the numbers 3 and

5. Since both are odd, we must add them to

9. So, the calculation becomes:

$9 + 3 = 12$

$9 + 5 = 14$. This won't work.

Let's try to make the answer to $9 + \square = \text{even}$.

Since 9 is odd, we need to add an odd number to get an even number. This can either be 3 or

5. We are left with either 3 and 4 or 5 and

4. If 3 is chosen, then $9 + 3 = 12$, an even number. Then we can choose 4 or 5 to add to

1

2. Since 4 is even, $12+4=16$. That leaves 5 left. We have $5*5=25$ which does not work.

If 5 is chosen, then $9 + 5 = 14$, an even number. Then we can choose 3 or 4 to add to

1

2. We need to add an even number to

1

2. $12+4=16$. Then we have 3 remaining. We have $5 * 3=15$ which does not work.

However, the problem requires an *even number* as the result of *each calculation*.

$5 \times 4 = 20$

$9 + 3 = 12$

That means that $12 + 5$ has to be even. But it equals

1

7. I will have to rethink this problem.

Consider the numbers again. 3, 4, and

5. Also, consider $12 + \square$. If we put 4, we get 16, which is even. So $12 + 4 = 16$.

Then we are left with 3 and

5. If $5 \times 3 = 15$, which is odd. Therefore, this will not work.

There seems to be no solution based on the constraints.

3. Final Answer

$5 \times 4 = 20$

$12 + 3 = 15$ (Should be

4. then $12+4=16$)

$9 + 5 = 14$ (Should be 3 or 5.)

$12 + 4 = 16$

$9+3=12$

$5 \times$ is wrong.

$5 \times 3 = 15$

$12 + 4 = 16$

$9 + 5 = 14$

There is no solution, as the expression $5 \times \square$ cannot be even, nor can all results be even.

Final Answer: No solution.