Types of Integers Study Guide for McGraw-Hill's Firefighter Exams

Odd and Even Numbers

Odd Numbers are numbers that when divided by 2 leave a remainder.

Example:–3,–1, 1, 3, … (3 ÷ 2 = 1 with 1 remainder)

Even Numbers are numbers that when divided by 2 leave no remainder. Zero is included.

Example: –4,–2, 0, 2, 4, … (4 ÷ 2 = 2)

Note: Division by zero is undefined and therefore has no meaning regardless of whether the dividend is odd or even.

Signed Numbers

Negative and positive numbers are signed numbers. A thermometer has a scale having both positive and negative numbers. The numbers run along a vertical number scale with positive numbers, indicated by the (+) sign, being above zero and negative numbers, indicated by the (–) sign, found below zero.

Absolute Value of a Number

The absolute value of a number is its value when the sign is not taken into consideration. The symbol for absolute value is | |.

Example: |7| = 7 |–7| = 7

Adding Negative Numbers

Adding a negative number is the equivalent of subtracting a positive number.

Conversely, adding a positive number to a negative number is the same as subtracting the negative number.

Example: –4 + (+8) = 4

To add two negative numbers (like signs), add their absolute values and prefix the sum with their common sign.

Example: –6 + (–9) = |6| + |9| = –15

Subtracting Negative Numbers

Subtracting a negative number from a positive number is the equivalent of adding a positive number.

Conversely, subtracting a positive number from a negative number is the equivalent of adding the absolute value of two negative numbers and prefixing the answer with a minus sign.

Example: –4 – (+2) = –6

Multiplying and Dividing Signed Numbers

When multiplying or dividing two signed numbers, the product (multiplication) or quotient (division) will be positive if the signs are the same. If the signs are different, however, the product or quotient will be negative.

Example: (–5) × (–7) = +35

Example: (–6) ÷ (–3) = +2

Example: (–5) × (7) = –35

Example: (–6) ÷ (3) = –2

When multiplying more than two numbers, the product is always a negative number if there are an odd number of negative factors. Conversely, if there is an even number of negative factors, the product is always a positive number.

Example: (2) (4) (–4) = –32

Example: (8) (2) (–3) (–4) = 192

The product of two or more numbers is always zero if any of the numbers is zero.

Example: (–9) (–3) (0) = 0

When multiplying numbers in exponential (a number being multiplied by itself) form, odd exponents (powers) of negative numbers result in a product that is negative, and even exponents of negative numbers have positive products.

Example: (–3)3 (to the third power) = –3 × –3 × –3 = –27

Example: (–6)2 (to the second power) = –6 × –6 = 36

Note: Division by zero is undefined and therefore has no meaning regardless of whether the dividend (numerator) is a positive or negative number.

Prime Numbers

Unique numbers whose sole factors are 1 and themselves are called prime numbers. All prime numbers except 2 are odd.

Example: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, …

To calculate the prime factorization of a number, divide it and all its factors until every remaining integer of the group is a prime number.

Example: 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3. The numbers 2 and 3 are both prime numbers.

Example: 12 = 2 × 6 = 2 × 2 × 3. The numbers 2 and 3 are both prime numbers.

To find the greatest common factor (GCF) of the two examples above, determine the intersection of prime numbers of the two prime factorizations. Because both prime factorizations contain (2 × 2 × 3), the GCF is 12.

Multiples of a number are its products with the natural numbers.

The multiples of 4 are therefore: 4, 8, 12, 16, and so on.

The least common multiple (LCM) is also found using prime factorization. The LCM is equal to the multiplication of each factor by the maximum number of times it appears in either number. Using the two prime factorization examples above, once again, the number 2 appears three times for the number 24 and two times for the number 12, and the number 3 appears one time for both 24 and 12. Therefore, the LCM of 24 and 12 is the product of 2 × 2 × 2 × 3 = 24.