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WHAT IS DIVISIBILITY??? DIVISIBILITY MEANS THAT A GIVEN NUMBER CAN BE DIVIDED WITHOUT A REMAINDER. ANY TIME THIS HAPPENS, THE NUMBERS WE DIVIDED BY.

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Presentation on theme: "WHAT IS DIVISIBILITY??? DIVISIBILITY MEANS THAT A GIVEN NUMBER CAN BE DIVIDED WITHOUT A REMAINDER. ANY TIME THIS HAPPENS, THE NUMBERS WE DIVIDED BY."— Presentation transcript:

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3 WHAT IS DIVISIBILITY??? DIVISIBILITY MEANS THAT A GIVEN NUMBER CAN BE DIVIDED WITHOUT A REMAINDER. ANY TIME THIS HAPPENS, THE NUMBERS WE DIVIDED BY ARE CALLED FACTORS. MULTIPLYING A GIVEN NUMBER BY ANY OTHER NUMBERS CREATES A LIST OF MULTIPLES. SINCE DIVISION AND MULTIPLYING ARE INVERSE OPERATIONS, FACTORS AND MULTIPLES ARE KIND OF LIKE OPPOSITES

4 DIVISIBILITY RULES NUMBER DIVISIBILITY RULE 1 ALL NUMBERS ARE DIVISIBLE BY 1 2 NUMBERS ENDING IN AN EVEN DIGIT ARE DIVISIBLE BY 2 3 NUMBERS WHOSE DIGITS HAVE A SUM WHICH IS DIVISIBLE BY 3 ARE DIVISIBLE BY 3 4 WHEN THE NUMBER FORMED BY THE LAST TWO DIGITS OF A NUMBER IS DIVISIBLE BY 4, THE ENTIRE NUMBER IS ALSO. 5 NUMBERS ENDING IN 5 OR 0 ARE DIVISIBLE BY 5 6 NUMBERS THAT ARE DIVISIBLE BY 2 AND 3 ARE ALSO DIVISIBLE BY 6 9 NUMBERS WHOSE DIGITS HAVE A SUM WHICH IS DIVISIBLE BY 9 ARE DIVISIBLE BY 9 10 NUMBERS ENDING IN 0 ARE DIVISIBLE BY 10

5 FACTORS VS MULTIPLES FACTORSMULTIPLES NUMBERS THAT DIVIDE INTO A GIVEN NUMBER LEAVING NO REMAINDER THE FIRST FACTOR OF ANY NUMBER IS 1 THE LAST FACTOR OF ANY NUMBER IS THE NUMBER ITSELF NUMBERS THAT MULTIPLY TOGETHER TO MAKE THE GIVEN NUMBER ARE CALLED FACTOR PAIRS NUMBERS CREATED BY MULTIPLYING A GIVEN NUMBER BY CONSECUTIVE COUNTING NUMBERS THE FIRST FACTOR OF ANY NUMBERS IS THE NUMBER ITSELF THERE IS NO LAST MULTIPLE, A GIVEN NUMBER HAS INFINITE MULTIPLES 1, 2, 3, 4, 6, 8, 12, , 48, 72, 96, 120 … FACTORSMULTIPLES (COME BEFORE THE NUMBER)(COME AFTER THE NUMBER)

6 FINDING THE LCM WHEN WE COMPARE TWO OR MORE NUMBERS, THE LEAST COMMON MULTIPLE IS THE FIRST MULTIPLE THAT APPEARS ON EACH LIST 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 41, 45 … 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60 … 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, … IN THIS EXAMPLE: LCM OF 3 AND 4 IS 12 LCM OF 3 AND 9 IS 9 LCM OF 3, 4, AND 9 IS 36 HINT: HINT: MAKE THE LIST OF MULTIPLES FOR THE LARGEST NUMBER AND SEE IF THE OTHER NUMBERS DIVIDE INTO IT EVENLY

7 FINDING THE GCF WHEN WE COMPARE TWO OR MORE NUMBERS, THE GREATEST COMMON FACTOR IS THE LARGEST FACTOR THAT APPEARS ON EACH LIST EACH FACTOR SHOULD HAVE A PARTNER THAT MULTIPLIES WITH IT TO FORM THE GIVEN NUMBER…THESE ARE CALLED FACTOR PAIRS. DRAWING LINES TO CONNECT FACTOR PAIRS FORMS A FACTOR RAINBOW…THIS IS HOW WE KNOW IF WE HAVE ALL THE FACTORS OF THE GIVEN NUMBER. 24: 1, 2, 3, 4, 6, 8, 12, 24 42: 1, 2, 3, 6, 7, 14, 21, 42 THE ONLY TIME A FACTOR WILL NOT HAVE A PARTNER IS WHEN THE GIVEN NUMBER IS A PERFECT SQUARE (LIKE 5 x 5 = 25)

8 PRIME AND COMPOSITE NUMBERS A NUMBER WHOSE ONLY FACTORS ARE ONE AND ITSELF IS CALLED A PRIME NUMBER. ALL OTHER NUMBERS, WITH MORE THAN TWO FACTORS, ARE COMPOSITE AND 1 ARE NEITHER PRIME NOR COMPOSITE 2 IS THE ONLY EVEN PRIME NUMBER ALL ODD NUMBERS ARE NOT PRIME!

9 PRIME FACTORIZATION BREAKING A NUMBER DOWN TO A PRODUCT OF ITS PRIME FACTORS IS CALLED PRIME FACTORIZATION. WE CREATE A FACTOR TREE TO MODEL THE WAY A NUMBER IS BROKEN DOWN TO PRIMES, EACH LEVEL OF BRANCHES SHOWS A FACTOR PAIR, ANY FACTORS THAT ARE NOT PRIME MUST BE BROKEN DOWN TO SMALLER FACTOR PAIRS UNTIL THE END OF EVERY BRANCH IS A PRIME NUMBER AND 7 ARE A FACTORS THAT HAVE A PRODUCT OF 70… 10 IS NOT PRIME, SO WE BREAK IT DOWN TO A FACTOR PAIR OF 5 AND 2 THE ENDS OF THE BRANCHES ARE PRIME: 5, 2, AND 7 SO YOU ARE FINISHED!!! WRITE THE PRODUCT IN DESCENDING ORDER (BIG NUMBERS TO SMALL NUMBERS)

10 USING PRIME FACTORIZATION TO FIND THE GCF THERE IS ONE 5 AND ONE 2 IN COMMON TO THE PRIME FACTORIZATION OF 40 AND 50, 5 x 2 = 10, SO THE GCF IS 10

11 LESSON 4 VOCABULARY REVIEW TERMDEFINITION DIVISIBILITY DETERMINATION OF THE ABILITY TO DIVIDE A GIVEN NUMBER WITHOUT LEAVING A REMAINDER (10 IS DIVISIBLE BY 1, 2, 5, AND 10) FACTOR A NUMBER WHICH DIVIDES A GIVEN NUMBER WITH NO REMAINDER … FOR ANY NUMBER, THE FIRST FACTOR IS ALWAYS 1 AND THE LAST FACTOR IS ALWAYS THE NUMBER ITSELF (THE FACTORS OF 10 ARE 1, 2, 5, AND 10) FACTOR PAIR TWO FACTORS OF A GIVEN NUMBER THAT MULTIPLY TOGETHER TO CREATE THAT NUMBER (THE FACTORS OF 20 ARE 1, 2,4, 5, 10 AND 20…4 AND 5 ARE A FACTOR PAIR BECAUSE 4 X 5=20) FACTOR RAINBOW DIAGRAM USED TO MAKE SURE NO FACTORS ARE MISSED IN DETERMINING ALL THE FACTORS OF A GIVEN NUMBER GCF GREATEST COMMON FACTOR…THE LARGEST NUMBER WHICH IS A FACTOR OF EACH IN A SET OF GIVEN NUMBERS (THE FACTORS OF 10 ARE 1, 2, 5 AND 10; THE FACTORS OF 20 ARE 1, 2, 4, 5, 10 AND 20 … SINCE 10 IS THE BIGGEST NUMBER THAT APPEARS ON BOTH LISTS, 10 IS THE GCF OF 10 AND 20) MULTIPLE A NUMBER CREATED BY MULTIPLYING A GIVEN NUMBER BY ANY COUNTING NUMBERS (THE FIRST FIVE MULTIPLES OF 3 ARE 3, 6, 9, 12, AND 15 BECAUSE 3X1=3, 3X2=6, 3X3=9, 3X4=12, AND 3x5=15

12 LESSON 4 VOCABULARY REVIEW TERMDEFINITION LCM LEAST COMMON MULTIPLE…THE SMALLEST NUMBER WHICH IS A MULTIPLE OF EACH IN A SET OF GIVEN NUMBERS (THE FIRST 5 MULTIPLES OF 4 ARE 4, 8, 12, 16, AND 20; THE FIRST 5 MULTIPLES OF 3 ARE 3, 6, 9, 12, AND 15 … SINCE 12 IS THE FIRST NUMBER THAT APPEARS ON BOTH LISTS, 12 IS THE LCM) PRIME ANY NUMBER WITH EXACTLY TWO FACTORS: 1 AND THE NUMBER ITSELF (5, 7, 11, AND 19 ARE SOME PRIME NUMBERS; 2 IS THE ONLY EVEN PRIME NUMBER) COMPOSITE ANY NUMBER WITH MORE THAN TWO FACTORS (THE FACTORS OF 10 ARE 1, 2, 5, AND 10 THEREFORE 10 IS A COMPOSITE NUMBER) PRIME FACTORIZATION TO BREAK A NUMBER DOWN TO A PRODUCT OF ONLY PRIME FACTORS; A FACTOR TREE IS USED TO ORGANIZE THESE FACTORS, AND THE FINAL SOLUTION SHOULD BE EXPRESSED IN EXPONENTIAL FORM (THE PRIME FACTORIZATION OF 75 IS 5X5X3, EXPRESSED AS 5 2 X3) FACTOR TREE DIAGRAM USED TO ORGANIZE THE PRIME FACTORS OF A GIVEN NUMBER EXPONENTIAL FORM REPEATED MULTIPLICATION OF A CONSTANT NUMBER IS RE-WRITTEN USING THE NUMBER ITSELF AS A BASE AND THE AMOUNT OF TIMES IT APPEARS AS THE EXPONENT (5 X 5 X 5 X 5 WOULD BE 5 4 IN EXPONENTIAL FORM)


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