Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Infinitesimal wikipedia , lookup
Positional notation wikipedia , lookup
List of prime numbers wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Large numbers wikipedia , lookup
Location arithmetic wikipedia , lookup
Factorization wikipedia , lookup
Elementary arithmetic wikipedia , lookup
• DEFINITION OF FACTORS AND MULTIPLES. • TESTS FOR DIVISIBILITY. • PRIME AND COMPOSITE NUMBERS • PRIME FACTORISATION. • LCM AND HCF. • RELATION BETWEEN LCM AND HCF. • FACTORS: A factor is a number that can divide another number without leaving a remainder. For eg 24 is divisible by 2,3,4,6,8,12 and 24 so all these are factors of 24 • MULTIPLES: Multiples of any numbers are the numbers which are exactly divisible by the number. • For eg Multiples of 4 are 4x1=4, 4x2=8, 4x3=12 • DIVISIBILITY RULES FOR 2,3,4,5,6,9,10 • 2 -A number is exactly divisible by 2,if the last digit is 0,2,4,6 or 8. eg 30, 648, 122 • 3 - A number is exactly divisible by 3 if the sum of its digits is exactly divisible by 3.eg 231 2+3+1=6 and 6/2=3. • 5 • 6 -A number is exactly divisible by 5 if the last digit is 0 or 5. eg 305, 500 - A number is exactly divisible by 6, if it is exactly divisible by both 2 and 3. eg 48 48/2=24 48/3= 16 • 9 - A number is exactly divisible by 9 if the sum of its digits is exactly divisible by 9 eg 153 1+5+3=9 and 9/9=1 • 10 - A number is exactly divisible by10 if the last digit is 0.eg 20, 30 • 1 is the smallest factor of every number. • A factor of any number is always less than or equal to that number. • Every number is a multiple of 1. • Multiples are infinite while factors are limited. • Every number is a multiple of itself. • PRIME NUMBER: A number greater than 1 and having only two factors , is known as prime number. Eg 2,3,5,7 • NOTE: There is no even prime number except 2. • COMPOSITE NUMBER: A number greater than 1 and having more than two factors , is known as composite number. Eg 4,6,8,10 • TWIN PRIMES: Two prime numbers with a difference of 2 are called TWIN PRIMES. Eg 3 and 5 (difference=2) • DEFINITION: When every factor of a number is a prime number ,the factorisation of that number is known as prime factorisation of the number . Example: 18 = 2 x 9 = 2x3x3 2 and 3 are the prime factors of 18. Divide to find the prime factors 2 2 2 2 48 24 12 6 3 So prime factorisation of 48 is 2x2x2x2x3 • DEFINITON: The greatest number that divides the numbers exactly is called the HCF of the number • Example: Let us take two composite numbers 18 and 24 Factors of 18 = 1,2,3,6,9,18 Factors of 24 = 1,2,3,4,6,8,12,24 Common factors = 1,2,3,6 6 is the highest common factor so HCF = 6 • HCF BY PRIME FACTORISATION. • HCF BY LONG DIVISION METHOD. • EXAMPLE: Find the HCF of 20 and 24 by prime factorisation method. Solution 24 12 6 3 1 20 =2x2x5 24 = 2x2x2x3 common factor =2x2 2 20 2 10 5 5 1 So HCF OF 20 AND 24 IS 4 2 2 2 3 • STEP 1: Divide the greater number by the smaller number and find the remainder. • STEP 2: Divide the smaller number or the divisor by the remainder. • STEP 3: Continue till you reach the last divisor. It is the HCF. • EXAMPLE: Find the HCF of 20 and 24 by long division method. 1 • SOLUTION: 20 24 20 5 4 20 20 0 Since last divisor is 4. Therefore HCF is 4 • DEFINITION: The least multiple ,which is common for two or more numbers is their LCM. • For example let us take two numbers 2 and 6. Multiples of 2 – 2,4,6,8,10,12,14,16 Multiples of 6 - 6, 12,18 Common multiples – 6,12 but 6<12 Hence 6 is the LCM of 2 and 6. METHODS FOR LCM MULTIPLE OR MEANING METHOD PRIME FACTORISATION METHOD COMMON DIVISION METHOD MULTIPLE METHOD – We have already seen in the definition of LCM. EXAMPLE: Find the LCM of 8 and 10 by common division method. 2 8, 10 Solution: 2 4, 5 2 2 ,5 5 1,5 1, 1 So LCM of 8 and 10 = 2 x 2 x 2 x 5 = 40 EXAMPLE :Find the LCM of 12 and 16 by prime factorisation method. • SOLUTION: LCM of 12 and 16 is 2 12 2 6 3 3 1 2 16 2 8 2 4 2 2 1 12 =2x2x3 Common factors are 2,2, and uncommon factors are 3,2,2 so multiply 2x2x3x2x2 = 48 16 =2x2x2x2 Therefore LCM = 48 • • • • PRODUCT OF TWO NUMBERS = LCM X HCF LCM = PRODUCT OF TWO NUMBERS / HCF HCF = PRODUCT OF TWO NUMBERS /LCM EXAMPLE: Find the LCM of two numbers whose product is 120 and their HCF= 2. Solution: LCM = PRODUCT OF TWO NUMBERS / HCF =120/2 =60 so LCM =60 EXAMPLE: Find the HCF of two numbers whose product is 375 and their LCM =75 Solution: HCF = PRODUCT OF TWO NUMBERS /LCM = 375/75 =5 So HCF = 5