# Prime Factorization Problems

Prime factorization is a way of writing a composite number as a product of prime numbers. Prime factorizations of different numbers are used in finding the GCF and hence the LCM of the numbers. Many real world problems which require to use GCF or LCM are solved using prime factorization of numbers.

## Finding prime factorization problems

Find the prime factorization of numbers 32 and 96.
Let us first write the prime factorization of 32.  32 is an even number, hence divisible by 2. Division of 32 by 2 yields the quotient 16. Hence 32 can be written as
32 = 2 x 16
Continuing the division process we get the series of results as,
16 = 2 x 8    so, 32 = 2 x 2 x 8
8 = 2 x 4 and     32 = 2 x 2 x 2 x 4
4 = 2 x 2  and the division cannot be continued as 2 is a prime number.
Hence the prime factorization of 32 is
32 = 2 x 2 x 2 x 2 x 2    or using exponents, 32 = 25.

Now let us take the number 96. 96 is divisible by both 2 and 3.  Dividing 96 by 3 we get the quotient as 32.
96 = 3 x 32
Since we have already found the prime factorization of 32, we can substitute that in the equation to get the prime factorization of 96 as
96 = 3 x 2 x 2 x 2 x 2 x 2  or using exponents 96 = 25 x 3.
This method is handy when we know the prime factorization of composite factors of a given number.  We need not repeat the division process  and the prime factorization can be done in quick steps.

## GCF Problems

The greatest common factor (GCF) or the greatest common division (GCD) of two or more numbers is the product of all the common factors of the numbers. To find the common factors, the prime factorization of each number is used.

There are many problem situations which require the GCF of certain numbers to be found. These problems are solved writing the prime factorization of numbers involved.

Example:
A combined class of 6th, 7th and 8th grade students was assembled for a Math skills development workshop. The organizers seated the students in the order of grades and seating same number of students in each row. If there are 48  6th graders,  40 7th graders and 32 8th graders present, how many students were seated in a row?

The number of students seated in a row should divide the number of students in each grade completely. This means we need to find the greatest common divisor of the numbers 48, 40 and 32.
So writing the prime factorization of these three numbers,
48 = 2 x 2 x 2 x 2 x 3
40 = 2 x 2 x 2 x 5
32 = 2 x 2 x 2 x 2 x 2
The greatest common factor is hence = 2 x 2 x 2 = 8.
8 students were seated in each row.

## LCM Problems

The lowest common multiple of two or more numbers can be found by listing the multiples of the numbers in ascending order and picking the first occurring common multiple as the LCM.  But this method would be difficult for larger numbers where the lists would be quite long. In such instances the prime factorization and the GCF of the numbers are used to find the LCM of the number.

Jane visits Aunt Mary once in 21 days and Uncle John once is 28 days.  Today she visited both of them.  When will she be again visiting both Aunt Mary and Uncle John?

The coincidence of the visits will occur at interval whose length is equal to the LCM of 21 and 28 days.
The prime factorization of the numbers 21 and 28 are,
21 = 3 x      and    28 = 2 x 2 x 7.
7 being the only common factor, the GCF  of 21 and 28 = 7
The LCM of the two numbers is got by multiplying the GCF with the factors left out in the prime factorization.
Hence the LCM of 21 and 28 = GCF x 3 x 2 x 2  = 7 x 12 = 84.
She needs to make the two visits on the same day after 84 days.