Arithmetic Factoring LCD LCM GCD
Notes:009p.1Arithmetic Factoring LCD LCM GCDRev004?2016 ldArithmetic Factoring LCD LCM GCD1. Simple Divisibility Rules:An integer is divisible by: 2 if it ends in 0, 2, 4, 6, or 8.3 if the sum of digits is divisible by 3.4 if the last 2 digits are divisible by 4.5 if it ends in 0 or 5.6 if it is divisible by 2 and by 3.8 if the last 3 digits are divisible by 8.9 if the sum of digits is divisible by 9.10 if it ends in 0.Class Divisibility examples 1:Check if the number is divisible by#234568910a.80b.135c.137d.231e.48Answers Class Divisibility examples 1:Check if the number is divisible by#234568910a.80?????b.135???c.137d.231?e.48?????80 ends in 0, so 2 is a factor8+0=8 and 8 is not divisible by 3, so 3 is not a factor80÷4=20 so 4 is a factor80 ends in 0, so 5 is a factor80 is not divisible by 3, so 6 is not a factor80÷8=10 so 8 is a factor8+0=8 and 8 is not divisible by 9, so 9 is not a factor2. Prime and composite numbers. An integer is a prime number if it is greater than 1 and is not divisible by any other positive integer than itself and one. If an integer is divisible by any other positive integer besides 1 and itself, then it is a composite number.Class Prime and Composite Number examples 2:Check if the number is prime and/or composite.#Prime?Composite?a.91b.127Answers Class Prime and Composite Number examples 2:Check if the number is prime and/or composite.#Prime?Composite?a.91?b.127?91=9.591÷2=45.591÷3=30.391÷4=22.7591÷5=18.291÷6=15.1691÷7=1391÷8=11.37591÷9=10.1127=11.26127÷2=63.5127÷3=42.3127÷4=31.75127÷5=25.4127÷6=21.16127÷7=18.142857127÷8=15.875127÷9=14.1127÷10=12.7127÷11=11.543. Factors. Factors of an integer are the integers you multiply together to get it. The factors are the integers that divide evenly into the integer (i.e., there is no remainder or number to the right of the decimal point). Remember that 2 negative integers are factors of a positive integer also. A negative integer has a positive and a negative integer as factors.4. Prime factors. Every whole number can be factored into the product of primes in only one way (ignoring order of the factors). Include all copies of the factor. E.g., 8=2?2?2If the number is:48 or less: try 2, 3, and 5.120 or less: try 2, 3, 5, and 7.220 or less: try 2, 3, 5, 7, 11, and 13.8. To find the prime factors of 96, keep dividing by the primes 2, 3, 5, and 7 since 96 is less than 120.96÷2=4848÷2=2424÷2=1212÷2=66÷2=3The prime factorization of 96 is2?2?2?2?2?3=25?3Class Example 3. List the prime factors of the ##Prime factorsa.350Answers Class Example 3. List the prime factors of the ##Prime factorsa.3502 · 5 · 5 · 7350÷2=175175÷2=87.5 no175÷3=58.3 no175÷5=3535÷5=75. Composite factors. To find the composite factors of a number, remember your multiplication tables. Start with the number itself and 1. Then keep going. Use the simple divisibility rules also.The composite factors of 12 are:12?1 ; 6?2 ; 4?3(side note: ACT math 050n has notes on perfect square factors, perfect cube factors, etc.)6. Using calculator to find composite factors. If you don’t know your multiplication tables, but have a calculator, to find the composite factors, take the square root of the number you are trying to factor, and divide the number by all numbers from 1 up to the square root, and see which ones divide evenly7. Example with calculator: Find the composite factors of 96.1. 96=9.8 so divide by 1 to 9.96÷1=96 96÷2=4896÷3=3296÷4=2496÷5=19.296÷6=1696÷7=13.7196÷8=1296÷9=10.67The composite factors of 96 are the ones that divide evenly:96 and 1; 48 and 2; 32 and 3; 24 and 4; 16 and 6; 12 and 8.Class Example 4. List all positive composite factor pairsList all positive composite factor pairs of 45Answers Class Example 4. List all positive composite factor pairsList all positive composite factor pairs of 4545÷1=45 so pair (1, 45)45÷3=15 so pair (3, 15)45÷5=9 so pair (5, 9)45÷2=22.5 no45÷4=11.25 noClass Example 5. List all composite factor pairsList all composite factor pairs of -48Answers Class Example 5. List all composite factor pairsList all composite factor pairs of -481 , -482 , -243 , -164 , -126 , -8-1 , 48-2 , 24-3 , 16-4 , 12-6 , 8-48÷5=-9.6 noClass Example 6. List all composite factor pairs (including negative pairs)List all composite factor pairs of 28Answers Class Example 6. List all positive composite factor pairs (including negative pairs)List all composite factor pairs of 281 , 282 , 144, 7- 1 , - 28-2, -14- 4 , - 710. Multiple. A multiple of an integer is the product of that integer and another integer.Multiples of 7 are:1?7=72?7=143?7=214?7=285?7=35Class Example 8. List the first 6 multiples of 8.Answers Class Example 8. List the first 6 multiples of 8.1 · 8 = 82 · 8 = 163 · 8 = 244 · 8 = 325 · 8 = 406 · 8 = 4811. Least Common Multiple (LCM). The LCM of two integers is the smallest positive integer that is divisible by both.To find this, make lists of the multiples of each number, and compare the lists to find the smallest multiple that is in both lists. The LCM can also be found by multiplying together the highest power of the prime factors of the numbers.Class Example 9. Find the least common multiple (LCM) of the numbersnumbersLCMa.14,63Answers Class Example 9. Find the least common multiple (LCM) of the numbersnumbersLCMb.14,6312614?1=1463?1=6314?2=2863?2=12614?3=4214?4=5614?5=7014?6=8414?7=9814?8=11214?9=12612. Least Common Denominator (LCD). The least common multiple of the denominators of 2 fractions is called the LCD.The LCD is often used when adding and subtracting fractions with unlike denominators.Class Example 10. Find the least common denominator (LCD) of78 and 512Answers Class Example 10. Find the least common denominator (LCD) of78 and 512241 · 8 = 82 · 8 = 163 · 8 = 241 · 12 = 122 · 12 = 243 · 12 = 369. Greatest Common Divisor (GCD) The GCD of two or more numbers is the largest positive integer that divides the numbers without a remainder.To find this, list the divisors (each of numbers in factor pairs) of each number from small to large, and find the largest in each list.The GCD is useful for reducing fractions to lowest terms.Class Example 7. Find the greatest common divisor (GCD) of the numbersnumbersGCDa.28,14Answers Class Example 7. Find the greatest common divisor (GCD) of the numbersnumbersGCDa.28,14141, 282, 144, 71, 142, 713. GCD (cont.). The GCD can also be found by multiplying together the common prime factors of each number.Find the GCD of 48 and 1848=2?2?2?2?318=2?3?3The GCD is 2?3=614. GCD (cont.). The GCD of numbers >0 can be found using the Euclidean algorithm:gcd (a,a)=a gcd(a,b)=gcd(a-b,b) where a and b are ordered so that a>bgcd(48,18)=gcd(48-18,18)=gcd(30,18)gcd(30,18)=gcd(30-18,18)=gcd(12,18)gcd(18,12)=gcd(18-12,12)=gcd(6,12)gcd(12,6)=gcd(12-6,6)=gcd(6,6)=6 ................
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