Question

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

Polynomial factorization
ANSWERED
asked 2021-05-21
Use the unique factorization theorem to write the following integers in standard factored form.
4116

Answers (1)

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 \(2^{2}\times 3\times 7^{3}\).
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