Greatest Common Factor & Least Common Multiple Section 8.2 Muller

Greatest Common Factor: A number or quantity that is a factor of each member of a group of numbers or quantities For Example: What is the GCF of 12 and 18? Factors of 12: 1, 2, 3, 4, 6,12 Factors of 18: 1,2,3,6,9,18

Least common multiple: -The smallest quantity that is divisible by two or more given quantities without a remainder. For Example: What is the LCM of 6 and 8? Multiples of 6: 6,12,18,24,30,36,42,48,60,66,72,78… Multiples of 8: 8,16,24,32,40,48,56,64,72,80…

*Using the slide method, calculate the GCF and LCM of 8800 and 10,000 -You repeatedly find common factors. -Write the common factors on the left. -Write the answer that results from dividing by the common factors on the right. -The “slide” stops when the resulting answers on the right no longer have any common factors except 1. -The GCF is then the product of the factors down the left-hand side of the slide -The LCM is the product of the factors down the left-hand side and the numbers in the last row of the slide. *Using the slide method, calculate the GCF and LCM of 8800 and 10,000 Step 1: 8800 10,000 10 880 1,000 = 88 100 Step 2: 8800 10,000 = 880 1,000 Step 3: Step 4: 8800 10,000 10 880 1,000 88 100 = 44 50 8800 10,000 10 880 1,000 88 100 2 44 50 = 22 25 *10 doesn’t go into 88 evenly causing the common factor (10) to change to 2. L C M G C F GFC=10 x 10 x 2 x 2=400 LCM=10 x 10 x 2 x 2 x 22 x 25 =220,000

Using gcf and with fractions: -To put a fraction in simplest form, we can divide the numerator and the denominator of the fraction by the GCF. First determine that the greatest common factors of 24 & 36 is 12. Because… 2 x 12 2 __ = _____ = __ 36 3 x 12 3 *For Example Put 24/36 in simplest form 24 __ 36 2 __ 3 or Keep simplifying the fraction until it can no longer be simplified: 24 12 x 2 12 6 x 2 6 2 x 3 2 __ = _____ = __ = _____ = __ = ____ = __ 36 18 x 2 18 9 x 2 9 3 x 3 3

Using lcm with fractions: We can use the LCM of the denominators of two fractions to add fractions. In order to add fractions, we must first find a common denominator. Although any common denominator will do, the least common denominator works best. For Example: Add 1 3 __ & __ 6 8 LCM of 6 & 8 = 24 1 3 1 x 4 3 x 3 4 9 13 __ + __ = ____ + ____ = __ + __ = __ 6 8 6 x4 8 x 3 24 24 24 Answer in simplest form Common Denominator

Practice problems: 9= 9,18,27,36,45 1. Calculate the LCM of 9 & 12 12= 12,24,36,48,60 1. Calculate the LCM of 9 & 12 2. Calculate the GCF of 36 & 63 36= 63= 1,2,3,4,6,9,12,18,36 1,3,7,9,21,63

3. Show how to use the slide method to find the GCF & LCM of 224 & 392 Practice problems 2: 3. Show how to use the slide method to find the GCF & LCM of 224 & 392 224 392 2 112 196 56 98 28 49 4 7 GCF: 2 x 2 x 2 x 7= 56 LCM: 2 x 2 x 2 x 7 x 4 x 7 = 1,568

Even & odd -A counting number is even if it is divisible by 2 -In general, if a counting number has a 0, 2, 4, 6, or 8 in the ones place = even -A counting number is odd if it is not divisible by 2 -In general, if a counting number has a 1, 3, 5, 7, or 9 in the ones place = odd *Is 237, 921 even or odd? Section 8.4

The result of an Even number multiplied by another even number? The result is always an even number. Because each group can be divided by 2 with none left over and the whole collection of blocks can be divided into groups of 2 with none left over. Therefore, an even number times an even number is always an even number.

The results of Adding odd and even numbers? The result will always be odd. The odd number of suns can be divided into groups of 2 with I left over The even suns can be divided into groups of 2 with none left over, When the sun collections are joined together, there will be groups of 2, and the 1 sun that was left over from the odd group of suns will still be left over ODD EVEN ODD

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