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4.2 FACTORS AND SIMPLEST FORM As we said before, when we SIMPLIFY a fraction, we wish to write its equivalent using the smallest numbers possible. This is called writing the fraction in LOWEST TERMS. The numerator and denominator should have no common factors (that can be divided out!) other than 1. What are some “shortcuts” to know if a number is divisible by: 2 3 4 5 6 8 9 10 Ex: Fraction Lowest Terms? Common Factor(s)? Simplified to Lowest Terms: 5 7 14 21 _ 8 16 Notice that we want to divide out the LARGEST COMMON FACTOR possible. If we don’t at first, we just KEEP DIVIDING until there are no more common factors (except 1). Ex: Reduce to lowest terms: _ 5400 18000 Again, notice that the SIGN always remains the SAME for equivalent fractions. We can also use PRIME FACTORIZATION to divide out all possible common prime factors at once. We would then multiply together the final result knowing that there are no more common factors. First we need to know what prime factorization is!! PRIME NUMBERS are whole numbers that have exactly TWO DISTINCT factors, ITSELF and 1. Ex: 2, 3, 5, 7, _____, ______, _____. (For those interested, see me for a way to generate a list.) COMPOSITE NUMBERS are numbers with at least one other factor BESIDE itself and 1. Ex: ____, ____, ____ Note that the number ONE is _____ a prime factor. Why? The number ZERO is _______ a prime number. Why? As a matter of fact, zero and one are NEITHER PRIME NOR COMPOSITE!! Now we can define PRIME FACTORIZATION of a number, it is factorization where every factor is a PRIME NUMBER. Ex: 8 = 2 4 is NOT the prime factorization of 8. 8 = 2 3 is NOT any factorization of 8. Write 8 = 222 (or 23 ) to write its PRIME factorization. Lets see how this helps us write fractions in lowest terms, then we will explore how to easily “break it down”. Ex: Write in lowest terms 180 220 = 22335 2 2 5 11 = 33 11 = 9 11 To find the prime factorization of a number we can either use DIVISION by prime numbers, or we can use FACTORING TREES. When we get to the end, every factor should be prime. Lets try a couple: 144 3850 To use prime factorization to write fractions in lowest terms (simplest form): 1) Write the prime factorization of both the Numerator and the Denominator 2) Divide out common factors and replace them with the number ONE (to hold the place in multiplication, recall 11 = 1, not 2!) ***Recall x3 means x x x 3) Multiply the remaining factors in the Numerator and Denominator. Ex: Reduce by using prime factorization: a) 18 36 b) 36 18 c) 96 288 d) 6 3x 5 e) 8b c 20abc Next we will look at EQUIVALENT FRACTIONS. These are fractions which represent the same number (point) on the number line. (Insert from old 4.1.) List equivalent fractions for ½. What do you notice? Two fractions are equivalent if: Do p. 236~ # 52, 62, 72, 94.