Transcript Multiple
Number Theory
• Divides: b divides a if there is a whole number
q such that a = qb.
• Factor: b will be a factor of a.
• Divisor: b is also a divisor of a.
• Multiple: a is a multiple of b.
Multiples
• Multiples of 2
2x0, 2x1, 2x2, 2x3, 2x4,………
0, 2, 4, 6, 8,……
Multiples of 3
3x0, 3x1, 3x2, 3x3,…..
0, 3, 6, 9,……..
Divisors or Factors
• Divisors or Factors of 6:
We need to find whole numbers b & q such that
6 = bq
• Using arrays:
1x 6, 2x3
• Rainbow method:
1,2,3,6
Prime & Composite
• Prime: a natural number that has exactly 2
different factors, namely 1 and itself is prime.
• Composite: a natural number that has more
than 2 different factors is composite.
• One is called a unit and is neither Prime nor
Composite
Primes
• Sieve of Eratosthenes
2
3
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Factor Trees
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180
600
675
360
Test for divisibility
A number N is divisible by
• 2 if the last digit is an even digit(0,2,4,6,8).
• 3 if the sum of the digits is divisible by 3.
• 5 if the last digit is a 0 or 5.
• 6 if it is divisible by 2 & 3.
• 9if the sum of the digits is divisible by 9.
• 10 if the last digit is 0.
Tell which of the following are
divisible by 2,3,5,6,9,or10
•
•
•
•
43,826
111,111
26,785
5,280
Divisiblity by 11
• If the difference between the sums of every
other digit is divisible by 11 then the number
is divisible by 11.
• 34,567
• 343,244
• 92,252,191,213
Greatest Common Factor (Divisor)
The Greatest Common Factor of two numbers
m & n will be the number d the divides both
m & n at the same time.
GCF(m,n) = d
GCF(18,45)= ?
18 = {1,2,3,6,9,18}
45 = {1,3,5,9,15,45}
F18 F45 ={1,3,9}
GCF(18,45) = 9
• GCF( 24,27) =
• GCF( 14,27) =
• GCF(110, 132) =
Using Cuisenaire Rods
• Pg 265 # 16
Least Common Multiple
Least Common Multiple: the smallest common
multiple of both m and n is called their Least
common Multiple, LCM(m,n)
LCM(9,15)=
M9 = {9,18,27,36,45,54,63,72, 81,90,...}
M15 = {15,30,45,60,75,90,105,…}
M9 M15 = { 45,90,…}
LCM(9,15) = 45
• Finding GCF & LCM by using Prime Power
Representation.
• Euclidean algorithm.
Homework
• Pg 241 # 5,6,8a,9 all,11,16,35-39
• Pg 253 # 1,5,6,7,8,9,23,24
• Pg 265 # 1,2,5,8all,9,17a,31-34