Question

# Use the unique factorization theorem to write the following integers in standard factored form. 4116

Polynomial factorization
Use the unique factorization theorem to write the following integers in standard factored form.
4116

2021-05-22
Step 1
Unique Factorization Theorem.
For every integer n>1, there exists a positive integer k and distinct prime numbers
$$p_{1},p_{2}, p_{3},....,p_{k}$$ and positive integers such that,
$$n=p_{1}^{e_{1}}p_{2}^{e_{2}}...p_{k}^{e_{k}}$$
Step 2
Perform prime factorization of the number.
4,116 can be divided by 2.
$$4116=2\times 2058$$
2058 can also be divided by 2.
$$4116 = 2\times 2058$$
$$=2\times 2\times 1029$$
$$=2^{2}\times 1029$$
1029 can be divided by 3.
$$4116=2^{2}\times 1029$$
$$=2^{2}\times 3\times 343$$
343 can be divided by 7.
$$4116=2^{2}\times 1029$$
$$=2^{2}\times 3\times 7\times 49$$
Again divide 49 by 7.
$$4116=2^{2}\times 3\times 7\times 49$$
$$=2^{2}\times 3\times 7\times 7\times 7$$
$$=2^{2}\times 3\times 7^{3}$$
Thus, the factorization of 4116 is $$2^{2}\times 3\times 7^{3}$$.