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What is the prime factorization of 1,260?

A. [tex]2 \times 2 \times 3 \times 3 \times 5 \times 7[/tex]
B. [tex]2 \times 3 \times 5 \times 6 \times 7[/tex]
C. [tex]4 \times 5 \times 7 \times 9[/tex]
D. [tex]2 \times 3 \times 5 \times 7[/tex]

Sagot :

GinnyAnsweridn
To determine the prime factorization of 1,260, let's analyze each option given and compare it to the correct prime factorization.

We know prime factorization involves expressing a number as a product of its prime factors only. Prime numbers are numbers greater than 1 that have no divisors besides 1 and themselves.

We start by reviewing each option:

### Option A: [tex]\(2 \times 2 \times 3 \times 3 \times 5 \times 7\)[/tex]
- Let's calculate the product:
[tex]\[ 2 \times 2 = 4 \][/tex]
[tex]\[ 4 \times 3 = 12 \][/tex]
[tex]\[ 12 \times 3 = 36 \][/tex]
[tex]\[ 36 \times 5 = 180 \][/tex]
[tex]\[ 180 \times 7 = 1260 \][/tex]
Therefore, this option gives us 1260.

### Option B: [tex]\(2 \times 3 \times 5 \times 6 \times 7\)[/tex]
- First, note that 6 is not a prime number (it is [tex]\(2 \times 3\)[/tex]).
- Let's calculate the product:
[tex]\[ 2 \times 3 = 6 \][/tex]
[tex]\[ 6 \times 5 = 30 \][/tex]
[tex]\[ 30 \times 6 = 180 \][/tex]
[tex]\[ 180 \times 7 = 1260 \][/tex]
Although the product is 1260, 6 is not a prime number. This cannot be the correct prime factorization.

### Option C: [tex]\(4 \times 5 \times 7 \times 9\)[/tex]
- Note that neither 4 nor 9 are prime numbers (4 is [tex]\(2 \times 2\)[/tex], 9 is [tex]\(3 \times 3\)[/tex]).
- Let's calculate the product anyway:
[tex]\[ 4 \times 5 = 20 \][/tex]
[tex]\[ 20 \times 7 = 140 \][/tex]
[tex]\[ 140 \times 9 = 1260 \][/tex]
The product is 1260, but both 4 and 9 are not prime numbers. This is not a prime factorization.

### Option D: [tex]\(2 \times 3 \times 5 \times 7\)[/tex]
- Let's calculate the product:
[tex]\[ 2 \times 3 = 6 \][/tex]
[tex]\[ 6 \times 5 = 30 \][/tex]
[tex]\[ 30 \times 7 = 210 \][/tex]
The product is only 210, which is not 1260.

Reviewing and calculating these options, we find that Option A, [tex]\(2 \times 2 \times 3 \times 3 \times 5 \times 7\)[/tex], correctly represents the prime factorization of 1,260, where all factors are prime and their product equals 1,260.

Thus, the best answer is:

A. [tex]\(2 \times 2 \times 3 \times 3 \times 5 \times 7\)[/tex]
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