Write 1,386 as the product of its prime factors.
A) $2 \times 3 \times 231$
B) $2 \times 3 \times 3 \times 7 \times 11$
C) $2 \times 693$
D) $2 \times 3 \times 7 \times 33$
Answer (2)
Find the prime factors of 1386 by dividing by prime numbers.
Divide 1386 by 2 to get 1386 = 2 × 693 .
Continue factoring: 693 = 3 × 231 = 3 × 3 × 77 = 3 × 3 × 7 × 11 .
The prime factorization is 2 × 3 × 3 × 7 × 11 .
Explanation
Understanding the Problem We are asked to express 1386 as a product of its prime factors. This means we need to break down 1386 into a multiplication of prime numbers. A prime number is a number greater than 1 that has only two factors: 1 and itself.
Dividing by 2 Let's start by dividing 1386 by the smallest prime number, 2: 1386 = 2 × 693
Dividing by 3 Now, let's factor 693. It is divisible by 3: 693 = 3 × 231
Dividing by 3 Again Next, factor 231. It is also divisible by 3: 231 = 3 × 77
Dividing by 7 Now, factor 77. It is divisible by 7: 77 = 7 × 11
Prime Factorization Since 7 and 11 are prime, we have found all the prime factors. So, the prime factorization of 1386 is: 1386 = 2 × 3 × 3 × 7 × 11
Matching the Answer Comparing our result with the given options, we see that option B matches our prime factorization.
Final Answer Therefore, the correct answer is B) 2 × 3 × 3 × 7 × 11 .
Examples
Prime factorization is a fundamental concept in number theory with many practical applications. For example, when creating secure codes and encryptions, prime numbers play a crucial role. Imagine you want to send a secret message to a friend. You can use prime factorization to create a key that only you and your friend know, making it very difficult for others to decode the message. This is similar to how online transactions and communications are secured every day!
The prime factorization of 1386 is 2 × 3 × 3 × 7 × 11 . This was obtained by successively dividing the number by the smallest prime numbers until only prime factors remained. The correct choice from the options provided is B) 2 × 3 × 3 × 7 × 11 .
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