1. DIVISIBILITY, FACTORS & MULTIPLES
DUFF’S MATH STUFF…
MAKING SENSE OF NUMBER SENSE
LESSON 4
DIVISIBILITY, FACTORS & MULTIPLES

2. 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

3. DIVISIBILITY RULES 1 2 3 4 5 6 9 10 NUMBER DIVISIBILITY RULE
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

4. (COME BEFORE THE NUMBER) (COME AFTER THE NUMBER)
FACTORS VS MULTIPLES
FACTORS
MULTIPLES
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
FACTORS MULTIPLES
(COME BEFORE THE NUMBER) (COME AFTER THE NUMBER)
1, 2, 3, 4, 6, 8, 12, , 48, 72, 96, 120 …

5. 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

6. 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)

7. 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.
0 AND 1 ARE NEITHER PRIME NOR COMPOSITE
2 IS THE ONLY EVEN PRIME NUMBER
ALL ODD NUMBERS ARE NOT PRIME!

8. WRITE THE PRODUCT IN DESCENDING ORDER (BIG NUMBERS TO SMALL NUMBERS)
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. 70 10 5 2 7
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
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)

9. USING PRIME FACTORIZATION TO FIND THE GCF40 10 5 2 4 50 10 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

10. LESSON 4 VOCABULARY REVIEW
TERM
DEFINITION
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

11. LESSON 4 VOCABULARY REVIEW
TERM
DEFINITION
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 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)