Introduction to least common multiple and greatest common divisor
Least common multiple
In mathematics, the least common multiple (lcm), of two rational numbers a and b is the smallest positive rational number that is an integer multiple of both a and b. Since it is a multiple, it can be divided by a and b without a remainder.
Greatest common divisor
In mathematics, the greatest common divisor (gcd), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. (Source: From Wikipedia).
Example Problems to Find the least Common Multiple and Greatest Common Divisor
Example problems to find the least common multiple of two numbers
Example 1
Find the least common multiple of 3 and 7.
Solution
To find the least common multiples of 3 and 7, first we have to find the multiples of 3 and 7.
Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56, 63, 70
Here, 21 is the lowest common number in the multiples of 3 and 7.
So, 21 is the least common multiple of 3 and 7.
Example 2
Find the least common multiple of 5, 11
Solution
The multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70.
The multiples of 11 = 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132.
Here, 55 is the lowest common number in the multiples of 5 and 11.
So, 55 is the least common multiple of 5 and 11.
Example problems to find the greatest common divisor of two numbers.
Example 1
Find the greatest common divisor of 54, and 40
Solution
The greatest common divisor of 54 and 40 can be found by the prime factors of the given numbers.
Prime factorization of 54 = 2 * 3 * 3 * 3
Prime factorization of 40 = 2 * 2 * 2 * 5
The common factors in the prime factorization is 2
So, 2 is the greatest common divisor of 54 and 40.
Example 2
Find the greatest common divisor of 21 and 35
Solution
Prime factorization of 21 = 3 * 7
Prime factorization of 35 = 5 * 7
Here, 7 is the common term.
So, 7 is the greatest common divisor of 21 and 35.
Between, if you have problem on these topics what are all prime numbers, please browse expert math related websites for more help on hard 6th grade math problems.
Practice Problems to Find the least Common Multiple and Greatest Common Divisor
Problem 1
Find the greatest common divisor of 3, 5, 7
Answer: 1
Problem 2
Find the greatest common divisor of 54, 27
Answer: 27
Problem 3
Find the least common multiple of 21 and 3
Answer: 21
Problem 4
Find the least common multiple of 51 and 16.
Answer: 816
Least common multiple
In mathematics, the least common multiple (lcm), of two rational numbers a and b is the smallest positive rational number that is an integer multiple of both a and b. Since it is a multiple, it can be divided by a and b without a remainder.
Greatest common divisor
In mathematics, the greatest common divisor (gcd), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. (Source: From Wikipedia).
Example Problems to Find the least Common Multiple and Greatest Common Divisor
Example problems to find the least common multiple of two numbers
Example 1
Find the least common multiple of 3 and 7.
Solution
To find the least common multiples of 3 and 7, first we have to find the multiples of 3 and 7.
Multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
Multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56, 63, 70
Here, 21 is the lowest common number in the multiples of 3 and 7.
So, 21 is the least common multiple of 3 and 7.
Example 2
Find the least common multiple of 5, 11
Solution
The multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70.
The multiples of 11 = 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132.
Here, 55 is the lowest common number in the multiples of 5 and 11.
So, 55 is the least common multiple of 5 and 11.
Example problems to find the greatest common divisor of two numbers.
Example 1
Find the greatest common divisor of 54, and 40
Solution
The greatest common divisor of 54 and 40 can be found by the prime factors of the given numbers.
Prime factorization of 54 = 2 * 3 * 3 * 3
Prime factorization of 40 = 2 * 2 * 2 * 5
The common factors in the prime factorization is 2
So, 2 is the greatest common divisor of 54 and 40.
Example 2
Find the greatest common divisor of 21 and 35
Solution
Prime factorization of 21 = 3 * 7
Prime factorization of 35 = 5 * 7
Here, 7 is the common term.
So, 7 is the greatest common divisor of 21 and 35.
Between, if you have problem on these topics what are all prime numbers, please browse expert math related websites for more help on hard 6th grade math problems.
Practice Problems to Find the least Common Multiple and Greatest Common Divisor
Problem 1
Find the greatest common divisor of 3, 5, 7
Answer: 1
Problem 2
Find the greatest common divisor of 54, 27
Answer: 27
Problem 3
Find the least common multiple of 21 and 3
Answer: 21
Problem 4
Find the least common multiple of 51 and 16.
Answer: 816
