Page 2 - Arithmetic Reasoning Study Guide for the ASVAB

Rules of Operations

There are a number of things you should remember to help you in arithmetic calculations, especially when it comes time to evaluate your answer for accuracy. With this knowledge, you’ll often be able to just look at an answer and determine whether it could be right, or is most definitely wrong.

Even and Odd numbers

Addition

Adding two even integers always produces an even integer. Adding two odd integers always produces an even integer. Adding an odd number to an even number always produces an odd number:

\(2 +2 = 4\)
\(3 + 5 = 8\)
\(3 + 6 = 9\)

Subtraction

The same property holds true with subtraction:

\(4 -2 = 2\) (even)
\(5 - 3 = 2\) (even)
\(6 -3 = 3\) (odd)

Multiplication

Multiplying two even integers together always produces an even integer:

\[8\cdot 6 = 48\]

Multiplying an even number by an odd number always produces an even number:

\[8\cdot 5 = 40\]

Multiplying an odd number by an odd number always produces an odd number:

\[7\cdot 5 = 35\]

Division

If the number is evenly divisible by its divisor, the even/odd property is the same as with multiplication. Otherwise there is a remainder, the quotient is a fraction, and so is neither even nor odd.

Positive and Negative Numbers

On a number line, values increase positively to the right of zero and negatively to the left. The further you go in either direction, the larger the numbers, but the value of the numbers only increases as you move right and decreases when moving left.

1-number-line.png

When working with positive and negative numbers, it helps to think of them as balances in a bank account. The more money you put into the account, the further right (positive) you will move on the number line. When you withdraw money, pay with a debit card, or write a check, you are subtracting money and you move left on the number line. If the balance reaches zero and you still remove money, you move into the negative numbers on the number line. Those numbers also become larger, but have a negative sign (-) and indicate you have “overdrawn”.

Note: If a number does not have a negative sign, we assume it is positive.

Addition and Subtraction

Subtracting a negative number is the same as adding a positive number:

\[2 - (-2) = 2 + 2 = 4\]

A negative number plus a negative number always produces a negative number:

\[-5 + (-9) = -14\]

A positive number added to a negative number produces a positive number, if the positive number is bigger than the absolute value of the (negative number), otherwise the result is negative:

\[-4 +7 = 3\]

Multiplication and Division

A positive integer times a positive integer always produces a positive integer:

\[5\cdot 5 = 25\]

A negative integer times a negative integer always produces a positive integer:

\[-5\cdot -5 = 25\]

But a positive integer times a negative integer always produces a negative integer:

\[-5\cdot 5 = -25\]

With division, the property for signs is the same as with multiplication, although, note that if the integer is not evenly divisible by its divisor, the result is a fraction, not an integer.

Absolute Value

The absolute value of a number is best thought of as its distance from 0 on the number line. We sometimes use this symbol when working with negative integers, placing it between two of these vertical line symbols: \(\vert\)

For example, the number 9 is a positive number equal to the absolute value of \(-9\): \(9 = \vert-9\vert\). When the absolute value is used with algebraic equations, this means there will be two answers.

For example, with \(\vert x-9\vert = 5\), x can be 14 or 4.

To define absolute value takes two equations and makes it a nice example of a piecewise function. Suppose you have the simple equation \(y=\vert x \vert\). If \(x\) is positive, then we just have \(y =x\), but if \(x\) is negative, we have to change it to positive by negating the negative.

The two different equations to define \(y=\vert x \vert\) are:

\[y = x \text{ if } x \gt 0\] \[\text{ and }\] \[y = -x \text{ if } x \lt 0\]

2-piece-wise-function.png

In the graph you can see that the piece on the right is linear and the piece on the left is linear, but the graph as a whole isn’t linear. That’s why one equation can’t do the job. We have graphed the first equation on the right and the second equation on the left.

To summarize, a piecewise function is one in which more than one equation is used to define the output of the function.

Additional Multiplication and Division Ideas

Multiplication is repeated addition: it is adding the number being multiplied a certain number of times. For example with the expression \(2\cdot 4\), we add 2 four times to itself like so:

\[2 + 2 + 2 + 2 = 8\]

You can see this is equal to the product we memorized in grade school multiplication tables.

Division is equal sharing of some number of things. In the simplest way, it can be thought of as sharing equally a certain number of things, with a smaller number of other things.

For example if we have four apples, and we want to share them among two people, we divide 4 by 2:

\[4\div 2 =2\]

Each person gets two apples.

Prime Numbers

All integers can be expressed as a product of the number one and itself, for instance: \(4 = 4\cdot 1 \text{ and }3 = 3\cdot 1\). Any integer is always evenly divisible by itself and 1. But only a prime number is evenly divisible only by itself and one; there are no other numbers that evenly divide a prime number.

The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

An integer that is not a prime number is called a composite number, and it is evenly divisible by other integers (not just one and itself).

For example, the number 12 is evenly divisible by 1, 2, 3, 4, 6, and 12.

Factors

The factors of an integer are all the numbers that evenly divide that integer. These numbers multiplied together in the right combination result in a product equal to this number. The number 12 has six factors that evenly divide it. So 12 can be expressed in the following ways:

\(12 = 12\cdot 1\)
\(12 = 6\cdot 2\)
\(12 = 4\cdot 3\)

Greatest Common Factor

Suppose two numbers, say \(30\) and \(24\), have factors in common. The largest of those is the greatest common factor.

The factors of \(30\) are \(1, 2, 3, 5, 6, 10, 15 \text{ and }30\).
The factors of \(24\) are \(1, 2, 3, 4, 6, 8, 12, \text{ and }24\).
The common factors are \(1, 2, 3, \text{ and }6\). We can see that \(6\) is the greatest common factor.

Prime Factorization

Prime factorization is the process of factoring a given number so that all of its factors are prime. For example, in the section above, we found all the factors of \(24\), but only \(2 \text{ and } 3\) are primes, so we’re going to have to use them multiple times to get \(24\). A good way to do that is by successive division until you reach \(1\):

\[24 \div 3 = 8\] \[8\div2 = 4\] \[4\div2=2\] \[2\div 2 = 1\]

We used \(3\) once and \(2\) three times, so the prime factorization is \(2 \cdot 2 \cdot 2 \cdot 3\). The product of those is \(24\).

Multiples

A multiple of an integer is a product of two or more integers. The number 12 is a multiple of all six of its factors, because those factors multiplied together in the right combination produce the number 12.

Least Common Multiple

Think of two numbers, say \(4\) and \(10\). Let’s look at the multiples of these numbers.

Multiples of \(4\) are \(4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44\)…
Multiples of \(10\) are \(10, 20, 30, 40 , 50\)…

We see two multiples that are common to both numbers, \(20\) and \(40\). The least of them (least means lowest) is \(20\), so \(20\) is the least common multiple. It comes in handy when you’re adding fractions and need to find a common denominator.

Remainders

The remainder refers to the quantity left over after dividing one number by another: with \(16\div 5\), 5 divides 16 three whole times, leaving a remainder of 1.

Divisibility Rules

Divisibility rules help us to identify what integers exactly divide a given number. Put another way, divisibility rules help us to find all the factors of a given number. Divisibility rules for numbers 2 through 10 are:

2—All even numbers are evenly divisible by 2. A number is even if its last digit is evenly divisible by 2.

3—A number is divisible by three if the sum of its digits is divisible by three. Example: 12 contains the digits 1 and 2, and 2 +1 =3. Three is evenly divisible by three.

4—If the last two digits of a number are evenly divisible by 4, the number is divisible by 4.

5—If the last digit is 0, or if the last digit is 5, the number is divisible by 5.

6—If the integer is divisible by 2 and 3, it must be evenly divisible by 6.

7—Multiply the last digit (the digit furthest to the right) by 2, then subtract this product from the number to the left of the last digit. If the result is 0 or divisible by 7, the original number is divisible by 7. If the result is not 0, and it is still not clear, this test can be repeated. For example with 343, twice the last digit is 6. Subtract this from 34 to get 28, which we can quickly see is divisible by 7.

8—If the last three digits are divisible by 8, the number is evenly divisible by 8.

9—Similar to the rule for 3: if the sum of the digits in a number are evenly divisible by 9, the number is evenly divisible by 9. Example: with 81 the sum of the digits is 9, which is evenly divisible by 9.

10—The last digit ends in 0.

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