Transcript FACTORS, MULTIPLES, & DIVISIBILITY

DIVISIBILITY,
FACTORS & MULTIPLES
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
DIVISIBILITY RULES
NUMBER
DIVISIBILITY RULE
1
2
3
ALL NUMBERS ARE DIVISIBLE BY 1
4
WHEN THE NUMBER FORMED BY THE LAST TWO DIGITS OF A
NUMBER IS DIVISIBLE BY 4, THE ENTIRE NUMBER IS ALSO.
5
6
NUMBERS ENDING IN 5 OR 0 ARE DIVISIBLE BY 5
9
NUMBERS WHOSE DIGITS HAVE A SUM WHICH IS DIVISIBLE
BY 9 ARE DIVISIBLE BY 9
10
NUMBERS ENDING IN AN EVEN DIGIT ARE DIVISIBLE BY 2
NUMBERS WHOSE DIGITS HAVE A SUM WHICH IS DIVISIBLE
BY 3 ARE DIVISIBLE BY 3
NUMBERS THAT ARE DIVISIBLE BY 2 AND 3 ARE ALSO
DIVISIBLE BY 6
NUMBERS ENDING IN 0 ARE DIVISIBLE BY 10
FACTORS VS MULTIPLES
FACTORS
MULTIPLES

NUMBERS THAT DIVIDE INTO A GIVEN
NUMBER LEAVING NO REMAINDER


THE FIRST FACTOR OF ANY NUMBER IS 1

NUMBERS THAT MULTIPLY TOGETHER TO
MAKE THE GIVEN NUMBER ARE CALLED
FACTOR PAIRS
THE LAST FACTOR OF ANY NUMBER IS THE
NUMBER ITSELF

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
FACTORS
(COME BEFORE THE NUMBER)
1, 2, 3, 4, 6, 8, 12, 24
MULTIPLES
(COME AFTER THE NUMBER)
24
24, 48, 72, 96, 120 …
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: MAKE THE LIST OF MULTIPLES FOR THE LARGEST NUMBER
AND SEE IF THE OTHER NUMBERS DIVIDE INTO IT EVENLY
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)
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.
1 2 3 4 5 6 7 8 9 10 11
12 13 14 15 16 17 18 19 20
21 22 24 25 26 27 28 29 30
0 AND 1 ARE NEITHER PRIME NOR COMPOSITE
2 IS THE ONLY EVEN PRIME NUMBER
ALL ODD NUMBERS ARE NOT PRIME!
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.
10 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
70
10
5
7
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)
USING PRIME FACTORIZATION
TO FIND THE GCF
50
40
10
5
10
4
2
2
2
5
5
2
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
LESSON 4 VOCABULARY REVIEW
TERM
DIVISIBILITY
FACTOR
FACTOR PAIR
FACTOR
RAINBOW
DEFINITION
DETERMINATION OF THE ABILITY TO DIVIDE A GIVEN NUMBER
WITHOUT LEAVING A REMAINDER (10 IS DIVISIBLE BY 1, 2, 5, AND 10)
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)
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)
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
LESSON 4 VOCABULARY REVIEW
TERM
LCM
PRIME
COMPOSITE
DEFINITION
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)
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)
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 52X3)
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 54 IN EXPONENTIAL
FORM)