Divisibility, Factors, and Multiples An Integer is said to have divisibility with another non-zero Integer if it can divide into the number and have a remainder of zero. Remember: Zero divided by any number is zero Any number divided by zero is undefined One Integer is a factor of another if it divides that Integer with a remainder of zero. A prime number is an Integer greater than 1 with exactly two factors: 1 and the number itself. Two is the only even prime number. A composite number is an Integer greater than 1 with more than two positive factors. The number 1 is neither prime nor composite. A multiple of a Natural number is the product of that number and another Natural number.
Divisibility Rules for Selected Numbers A number is divisible by... if... 2 It is even (ends in 0, 2, 4, 6, or 8) 5 It ends in 0 or 5. 10 It ends in 0. 3 9 The sum of the digits are divisible by 3. The sum of the digits are divisible by 9. 6 It is divisible by 2 and 3. 4 The last two digits are divisible by 4, including 00. Double the last digit and subtract from the rest of the number. The result must be either 0, or divisible by 7.
The factors of a number can be listed in ascending order. Two factors can be multiplied together to obtain the number that they are a factor of. Ex: Factors of 12: {1, 2, 3, 4, 6, 12} List the factors of 36: Factor pairs of a number can be arranged in a rectangular formation using grid-squares.
Exponents A power has two parts: a base and an exponent. The exponent tells how many times to multiply the base by itself. a x
Prime Factorization and Greatest Common Factor Writing a composite number as the product of its prime factors shows the prime factorization of a number. Each number has its own distinct prime factorization. Think of this as a number's DNA blueprint!!! A factor tree is often used to help write the prime factorization of a number. Circle the prime factors as you move down the tree! The prime factorization is written in the ascending order of its bases. Exponents are used to show repeated factors. 825 72
Above is the prime factorization of a number. Write down three factors of the number that are each greater than 100.
HW: Choose 4 perfect squares and write their prime factorizations: A number is a perfect square if its prime factorization contains only EVEN exponents.
Finding the Greatest Common Factor (GCF) The GCF is the largest shared factor(s) between two or more expressions. 1a) 40 and 60 Using prime factorization to find the Greatest Common Factor between two or more expressions: 40 = 60 = 1b) 42 and 36
Find the GCF between 99 and 28. When two or more expressions do not have any common factors, their Greatest Common Factor is ONE. The expressions are said to be Relatively Prime.
Example: Find the GCF between 36a 2 g 12, 48g 3, and 40a 4 g 10 Ex: Find the GCF between 99w 4 g 2 and 28k 23 r 2
NOTE: For this example there are NO common factors between the expressions... Ex: Find the GCF between 99w 4 g 2 and 28k 23 r 2 When two or more expressions do not have any common factors, their Greatest Common Factor is ONE. The expressions are said to be Relatively Prime.
Least/Lowest Common Multiple A multiple is the product of a Natural number and another Natural number. The Least/Lowest Common Multiple is the smallest shared multiple between two or more expressions. The LCM will be divisible by each expression. To find the Lowest Common Multiple (LCM) between two expressions: A) 6 and 16 B) 6a 2 b and 32a 3
Find the Greatest Common Factor and Lowest Common Multiple of the following two expressions: October 17, 2014
If a number is a perfect square, the exponents of each base in its prime factorization will be EVEN.
Simplifying Fractions A fraction is in simplest form when the numerator and denominator have no common factors other than 1. You can divide the numerator and denominator by the Greatest Common Factor in order to simplify. Remember: A number (or variable) divided by itself is equal to ONE!!!!!