Add the ones digits:

\(\displaystyle 1+3+5=9\)

Since there is no tens digit to carry over, proceed to add the tens digits:

\(\displaystyle 1+1+1=3\)

The answer is \(\displaystyle 39\).

If there are \(\displaystyle 257\) students taking Greek, then there are \(\displaystyle 257+15\) or \(\displaystyle 272\) students taking Latin. However, the question asks how many total students there are in the school, so you must add these two values together to get:

\(\displaystyle 257 + 272\) or \(\displaystyle 529\) total students.

Rewrite the question in a mathematical expression.

\(\displaystyle 13+19\)

Add the ones digit.

\(\displaystyle 3+9 =12\)

Since this number is larger than \(\displaystyle 10\), carry over the \(\displaystyle 1\) in tens digit when adding the next term.

Add the tens digit with the carry over.

\(\displaystyle 1+1 + (1)=3\)

Combine the tens digit and the ones digit. The answer is \(\displaystyle 32\).

In order to get an odd result from an addition, we must have one odd and one even number, thus you know from the first point about \(\displaystyle a+b\) that only one of the two values is odd. Now, to get an even result, you can have two evens or two odds. So, let's presume that \(\displaystyle b+c\) has two odd values, this means that \(\displaystyle a\) must be even. Thus, you have:

\(\displaystyle a:even, b:odd,c:odd\)

Now, if we presume that \(\displaystyle b+c\) has two even values, we must then know that \(\displaystyle a\) is odd.  Thus, we have:

\(\displaystyle a:odd, b:even,c:even\)

First of all, you can eliminate the two answers that say that a given value is positive or negative. This cannot be told from our data. Next, it cannot be the case that \(\displaystyle a+c\) is even. It will always be odd (hence, the correct answer is this). Finally, it cannot be that \(\displaystyle a+b+c\) is even always. In the second case above, you will have two even numbers added together, given you an even. Then, you will add in an odd, giving you an odd.

There are two ways to solve this problem. First you can do so algebraically by dividing both sides by 13:

\(\displaystyle \frac{351}{13} = x\)

\(\displaystyle x = 27\)

But, there is another way, which if you understand odd numbers, is even faster. Of all the answers above, only one is odd. You know, given the equation, that \(\displaystyle x\) must be odd--any odd number multiplied by an odd number will yeild an odd number.  If you multiply an odd number by an even number, you will get an even number. 

There are two ways to approach this problem:

1. Use the rule that states that any two odd numbers multiplied together will yield another odd number. 

Using this rule, only one answer is an odd number (29) which will yield another odd number (493) when multiplied by the given odd number (17).

2. Solve algebraically:

\(\displaystyle 17x=493\)

\(\displaystyle \frac{493}{17}=29=x\)

There are two ways to approach this problem:

1. Use the rule that states that any two odd numbers multiplied together will yield another odd number. 

Using this rule, only one answer is an odd number (85) which will yield another odd number (7,735) when multiplied by the given odd number (91).

2. Solve algebraically:

\(\displaystyle 91x=7,735\)

\(\displaystyle \frac{7,735}{91}=85=x\)

With many questions like this, it might be easier to plug in numbers rather than dealing with theoretical variables.  However, given that this question asks for the expression that is not always even or odd but only not necessarily odd, the theoretical route might be our only choice.

Therefore, our best approach is to simply analyze each answer choice.

\(\displaystyle m^2n\):  Since \(\displaystyle m\) is odd, \(\displaystyle m^2\) is also odd, since and odd number multiplied by an odd number yields an odd product.  Since \(\displaystyle n\) is also odd, multiplying it by \(\displaystyle m^2\) will again yield an odd product, so this expression is always odd.

\(\displaystyle m-2n\):  Since \(\displaystyle n\) is odd, multiplying it by 2 will yield an even number.  Subtracting this number from \(\displaystyle m\) will also give an odd result, since an odd number minus an even number gives an odd number.  Therefore, this answer is also always odd.

\(\displaystyle mn\):  Since both numbers are odd, their product will also always be odd.

\(\displaystyle 2m-n\):  Since \(\displaystyle m\) is odd, multiplying it by 2 will give an even number.  Since \(\displaystyle n\) is odd, subtracting it from our even number will give an odd number, since an even number minus and odd number is always odd.  Therefore, this answer will always be odd.

\(\displaystyle \frac{m+n}{2}\):   Since both numbers are odd, there sum will be even.  However, dividing an even number by another even number (2 in our case) does not always produce an even or an odd number.  For example, 5 and 7 are both odd.  Their sum, 12, is even.  Dividing by 2 gives 6, an even number.  However, 5 and 9 are also both odd.  Their sum, 14, is even, but dividing by 2 gives 7, an odd number.  Therefore, this expression isn't necessarily always odd or always even, and is therefore our answer.

To find the answer to this question, calculate the total jelly beans for each person:

Portia: \(\displaystyle 3\) * <Theodore's count of jelly beans>, which is \(\displaystyle 3 * 15\) or \(\displaystyle 45\)

Harvey: \(\displaystyle 5\) * <Portia's count of jelly beans>, which is \(\displaystyle 5*45\) or \(\displaystyle 225\)

So, the total is:

\(\displaystyle 15+45+225=285\)

(Do not forget that you need those original \(\displaystyle 15\) for Theodore!)

Rewrite the product into the expression \(\displaystyle 13\cdot 7\), and multiply the ones digit of both numbers.

\(\displaystyle 3(7)=21\)

The ones digit of the final answer is the ones digit from the multiplication of 3 and 7 thus the 1.  The tens digit, 2, will be carried over to the next calculation.

Multiply the tens digit of 13 to 7 and add the carry over.

\(\displaystyle 7(1)+2=9\) 

The value of 9 is the tens digit of the final answer.  

Combine the tens digit with the ones digit.  The answer is \(\displaystyle 91\).