2. The method of finding the greatest common factor of two numbers: (1) enumeration method; (2) screening method; Find the smaller factor of two numbers first, then circle the factor of another number and see which is the largest; (3) factorization method (4) short division. Take the common prime factor as the divisor, remove these two numbers from small to large, until the quotient of these two numbers is a prime number, and then multiply all the divisors, which is the greatest common factor of these two numbers;
For example, 18 and 27 divide these two numbers by the minimum prime factor 3 of 18 and 27 to see if the quotient of these two numbers is prime; Not a prime number, divide it down until the quotient is a prime number, and then multiply it by the divisor to get the greatest common factor of 18 and 27. The greatest common factor of 18 and 27 is 3×3=9.
Find the greatest common factor of each group of numbers in the following example. What did you find?
3 and 9 5 and 25 8 and 9 4 and 17
(1) Find the greatest common factor of each group number.
The greatest common factor of 3 and 9 is 3; The greatest common factor of 5 and 25 is 5; The greatest common factor of 8 and 9 is1; The greatest common factor of 4 and 17 is 1.
(2) Observe the characteristics of these two numbers in each group.
3 and 9; 3 is a multiple of 9, and 9 is a multiple of 3; 5 and 25 are factors of 25, and 25 is a multiple of 5. (The greatest common factor is a small number) 8 and 9; 8 and 9 are prime numbers with a common factor of only1; 4 and 17: 4 and 17 are prime numbers, and the common factor is only 1. (The greatest common factor is 1).
Summary: (1) When two numbers are multiples, the smaller number is the greatest common factor of these two numbers. (2) The greatest common factor of the coprime of two numbers is 1.
Least common multiple:
The method of finding the least common multiple of two numbers: (1) enumeration method; (2) screening method; Find multiples of two larger numbers and circle multiples of smaller numbers in descending order. The first circle is their least common multiple. (3) factorization method: (4) short division; Divide the prime factors of two numbers * * * as divisors in the order from small to large until the quotient prime number is obtained (if it is three numbers, until the obtained quotient is a paired prime number), and then multiply the quotients obtained by all the divisors to get the least common multiple of these two numbers.
For example, what is the least common multiple of 6 and 8?
Method analysis: Find the same prime factor 2 of 6 and 8, divide 6 and 8 by 2 to see if their quotient is prime, if it is, there is no need to divide it, and then connect the divisor 2 with the two obtained to get the least common multiple of 6 and 8. The least common multiple of 6 and 8 is 2× 3× 4 = 28.
Find the least common multiple of each group. What did you find?
12 and 36 3 and 1 1 8 and 9 5 and 25.
The least common multiple of 12 and 36 is 36; The minimum common multiple of 6× 6 = 36 ∴ 3 and 1 1 is 33; The least common multiple of 3× 1 1 = 338 and 9 is 72; The least common multiple of 5 and 25 is 25.
(2) Observe the characteristics of two numbers in each group. 5 and 25, 12 and 36 are multiples; Every two numbers in the groups 3 and 1 1, 8 and 9 are prime numbers.
(3) Find the relationship between the least common multiple of each group number and these two numbers.
The least common multiple of 12 and 36 is the larger of the two numbers, as are 5 and 25; The least common multiple of 8 and 9 is the product of these two numbers, so are 3 and 1 1.
Inductive summary
(! ) two numbers, if the larger number is a multiple of the smaller number, then the larger number is the least common multiple of these two numbers.
(2) If two numbers are prime numbers, then the product of these two numbers is their least common multiple.
Whether seeking the greatest common factor or the least common multiple, short division is the most suitable.