# Page 2 Arithmetic Reasoning Study Guide for the ASVAB

### Other Rules of Operation

#### Even and Odd numbers

Adding two even integers always produces an even integer.
Adding two odd integers always produces an even integer.
Adding an odd number with 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.

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

Multiplying an odd number by an odd number always produces an odd number.

##### 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:

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

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

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:

##### Multiplication and Division:

A positive integer times a positive integer always produces a positive integer.

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

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

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.

### Multiplication/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:

$2 + 2 + 2 + 2 = 8$
which you can see is equal to the product we memorized in our 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 2 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. $4 = 4\cdot 1\;and\;3 = 3\cdot 1$. All integers are 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 6 factors that evenly divide it. So 12 can be expressed in the following ways:

$12 = 12\cdot 1$
$12 = 6\cdot 2$
$12 = 4\cdot 3$

#### 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.

#### Remainders

This term refers to the quantity left over after dividing one number by another: with $16\div 5$, five 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 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