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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.

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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,……..

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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

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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

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Primes Sieve of Eratosthenes 2345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626367656667686970 71727374757677787980 81828384858687888990 919293949596979899100

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Factor Trees 180 600 675 360

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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.

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Tell which of the following are divisible by 2,3,5,6,9,or10 43,826 111,111 26,785 5,280

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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

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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} F 18 F 45 ={1,3,9} GCF(18,45) = 9

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GCF( 24,27) = GCF( 14,27) = GCF(110, 132) =

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Using Cuisenaire Rods Pg 265 # 16

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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)= M 9 = {9,18,27,36,45,54,63,72, 81,90,...} M 15 = {15,30,45,60,75,90,105,…} M 9 M 15 = { 45,90,…} LCM(9,15) = 45

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Finding GCF & LCM by using Prime Power Representation. Euclidean algorithm.

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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

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