in this tutorial we will talk about least common multiple or shortly lcm of numbers at the end of this tutorial you will be able to find least common multiple of any two three or several numbers let's get started a list common multiple is of several numbers is a multiple of each of them smallest multiple of each of them once again a common multiple of several numbers is a multiple of each of them for example 9 15 and 10 have a number 180 as common multiple because each of these numbers 9 15 and 10 divides 180 but beside 180 number 90 is also common multiple from these two numbers number 90 is the smallest number and this is the smallest number that numbers 9 15 and 10 divides divide that's why this 90 is called uh least common multiple of these numbers and we write lcm least common multiple of 9 15 and 10 is equal to 90. for small numbers lcm can be found by simple inspection but for big numbers lcm is equal to the multiplication of each factor in prime number factorization that once appear in any factorization with maximum exponent on each factorization to make it clear let's give examples example number one example let's find least common multiple lcm for numbers 252 441 and 1080. using prime number factorization we write factorization for each our numbers for each of our number for first number it will be 2 to the power of 2 times 3 to the power of 2 times 7.
441 is equal to 3 to the power of 2 times 7 to the power of 2 and 1080 is equal to 2 to the power of 3 and 3 to the power [Music] 3 times 5. so lcm list common multiple will be equal to we see that we have at least number two at least at one factorization and the biggest exponent in all factorization in each factorization is 3 so we write 2 to the power of 3 times uh three we have um in each factorization and the maximum and the biggest greatest exponent is three so we write 3 to the power of 3 times 7 because 7 here and here and the power of 7 is 2 times only 5 left so this is lcm of our numbers and if we multiply and write this number it will be equal to write it here above it will be equal to 52 and nine hundred and twenty so this number fifty two thousand and nine hundred and twenty is least common multiple each of our number 252 441 and 1080 each of this number divide 52 920 and this 52 and 920 the smallest number that three of our numbers divide uh this is the lcm the least common multiple let's solve let's give another example lcm of numbers 234 1080 8100 using prime number factorization we write factorization for each our number it will be 2 times 3 3 to the power 2 times 13. second number 1080 will be equal to two to the power of three times three to the power three times five eight thousand and one hundred will be equal to two to the power of two times three to the power four times five to the power two and least common multiple in this case will be equal to um let's write here equal to we have 2 here here and here and maximum exponent is three sorry maximum exponent is three that's why we write two to the power three times three we have number three we have here here and here and maximum exponent is four we write three to the power to power 4 times times 13 times 5 to power to the power of 2.
