# Prime Factorization and Number of Factors Algorithm (3rd through 7th grade)

The logical extension to finding all of the factors of a number using the double/half method is to dissect numbers into their prime factorization. Every whole number greater than one can be factored into prime numbers multiplied by other prime numbers. For example, 12 = 2x2x3 or we say 2^2 x 3. Remember that 3 can be represented as 3^1 as well but we use the simplified version of just 3.

I taught them the tree branch method of prime factorization where each pair of branches is the factor pair of the number above it. Please see the attached pdf for a detailed explanation.

The children were challenged to do the prime factorization of several numbers in the 2 and 3 digits and then I challenged them to go into 4 and 5 digits. They need simple short division to generate more difficult factors.

For the 4-7th graders, I challenged them to use the prime factorization to find the total number of factors in the number. The algorithm for this is to add one to each exponent in the prime factorization and then multiply all of these results. For example, in the number 180 has the prime factorization of 2^2 x 3^3 x 5. Since the exponents of these primes are 2, 3, and 1 respectively, we add one to each so, 3, 4, and 2. Then we multiply 3 x 4 x 2 to get 24 and we know that 180 has exactly 24 distinct factors. They can now use the double half method to check this result.

AttachmentSize
Prime_Factorization_Tree_Branch.pdf518.29 KB
Factor_Algorithm_from_prime_factorization.pdf222.17 KB