Divide the absolute values to get the magnitude of the answer.

\(\displaystyle 48\div16=3\)

Remember the following rules:

Positive divided by negative is negative.

Positive divided by positive is positive.

Negative divided by negative is positive.

Since we are dealing with two negatives, our answer will be positive.

\(\displaystyle -48\div-16=3\)

This is a classic order of operations question, and if you are not careful, you can end up with the wrong answer!  

Remember, the order of operations says that you have to go in the following order of operations: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction (also known as PEMDAS).  In this equation, you will start with the parentheses.  In the parentheses, we have 

\(\displaystyle (16\div2^{2})\).

But within the parentheses, you still need to follow PEMDAS.  First, we will solve the exponent, and the square of 2 is 4.  Then, we'll divide 16 by 4, which gives us 4, so we can rewrite our original equation as

\(\displaystyle -10+13-12\div4\).

We can now divide \(\displaystyle 4\) into \(\displaystyle -12\), which gives us

\(\displaystyle -10+13-3\). 

The last step is to add and subtract the numbers above, paying careful attention to negative signs.  In the end, we end up with \(\displaystyle 0\) because \(\displaystyle -10\) added to \(\displaystyle 13\) equals \(\displaystyle 3\), and \(\displaystyle 3\) minus \(\displaystyle 3\) equals \(\displaystyle 0\). 

For a number A to be divisible by another number B, A must share all of the prime factors of B.  For example, 100 is divisible by 10 because the prime factors of 10 (5 and 2) are found in the prime factors of 100 (2, 2, 5, and 5).  

In this problem, we have a number A that is divisible by 9 and 12.  First, find the prime factors of 9 and 12.  The factors of 9 are (3, 3).  The factors of 12 are (2,2,3).  So, to be divisible by 9 and 12, the number A must have the factors (2,2,3,3).  Multiply those together, and we get 2*2*3*3= 36.

So, A must be divisible by 36.  It could be divisible by the other answer choices, but since the question asked which choice must be right, we can only choose 36.

One way to do this:

Divide 100 by 6. The remainder will be the last digit.

\(\displaystyle 100 \div 6 = 16 \textrm{ R } 4\)

Now divide the quotient by 6. This remainder will be the second-to-last digit.

\(\displaystyle 16 \div 6 = 2 \textrm{ R } 4\)

The quotient is less than 6, so it is the first digit. The base-six equivalent of 100 is \(\displaystyle 244_{\textrm{six}}\).

All of these choices are equivalent to the correct answer. But only \(\displaystyle 2.5 \times 10^{10}\) is in scientific notation.

Write the factors of the two numbers.

\(\displaystyle \frac{32}{84}= \frac{4\times 8 }{4\times 21}= \frac{8}{21}\)

If a number is divisble by some number \(\displaystyle x\), then it is also divisible by all of its factors. Hence, if a number is divisible by \(\displaystyle 12\), it is also divisible by \(\displaystyle 1, 2, 3, 4\), and \(\displaystyle 6\). Thus, I and II are both true.

Rewrite the expression in fraction form.

\(\displaystyle 126 \div 36= \frac{126}{36}\)

Write out the common factors of both the numerator and denominator.

\(\displaystyle \frac{126}{36} = \frac{6\times 21}{6\times 6} =\frac{6\times 3\times 7}{6\times 3 \times 2}\)

We can then see that the fraction can be reduced to \(\displaystyle \frac{7}{2}\) without having to conduct long division.

The answer is \(\displaystyle \frac{7}{2}\).

Rewrite the expression as a fraction.

\(\displaystyle \frac{48}{14}\)

Write out the multiples of the numerator and denominator.

\(\displaystyle \frac{48}{14} =\frac{2 \times 3\times 8}{2 \times 7}\)

The twos can be cancelled.  Simplify what remains of the fraction.

\(\displaystyle \frac{3\times 8}{7}= \frac{24}{7}\)

The answer is:  \(\displaystyle \frac{24}{7}\)

Rewrite the expression as a fraction.

\(\displaystyle 38 \div 14 = \frac{38}{14}\)

Rewrite the fraction by common factors of two.

\(\displaystyle \frac{38}{14} = \frac{2 \times 19}{2\times 7}\)

Divide and cancel the twos, which will become one.

The answer is:  \(\displaystyle \frac{19}{7}\)