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Least common multiple

SA I write this post because I failed yesterday to solve the topcoder TCO 250 problem , it was about find the LCM -Least Common multiple -simply it is smallest number which accept the division on -non zero- numbers .

how we calculate it? there exist two ways lets discuss both:
for example 3,4,6
first way find the multiple of all numbers 2,3,4...etc.
Multiples of 2: 4,6,8,10,12,14,16.
Multiples of 3: 6,9,12,15.
Multiples of 4: 8,12,16,20.

as you see 12 is the smallest common number which appear on all numbers .

second way factor each number into it prime component and do the following steps :
1-Count the number of times each prime number appears in each of the factorizations.

2-For each prime number, take the largest of these counts.

3-Write down that prime number as many times as you counted for it in step 2.

4-The least common multiple is the product of all the prime numbers written down.

example 5,6,15.

Factor into primes

Prime factorization of 5 is 5

Prime factorization of 6 is 2 x 3

Prime factorization of 15 is 3 x 5

Notice that the different primes are 2, 3 and 5.

Now, we do Step #1 - Count the number of times each prime number appears in each of the factorizations...

The count of primes in 5 is one 5

The count of primes in 6 is one 2 and one 3

The count of primes in 15 is one 3 and one 5.

Step #2 - For each prime number, take the largest of these counts. So we have...

The largest count of 2s is one

The largest count of 3s is one

The largest count of 5s is one

Step #3 - Since we now know the count of each prime number, you simply - write down that prime number as many times as you counted for it in step 2.

Here they are...

2, 3, 5

Step #4 - The least common multiple is the product of all the prime numbers written down.

2 x 3 x 5 = 30

Therefore, the least common multiple of 5, 6 and 15 is 30.

So there you have it. A quick and easy method for finding least common multiples.

reference see the Link

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