Let's factorize the numbers 315 and 395 into their prime factors and then find their product.
Step 1: Prime Factorization of 315
1. [tex]\( 315 \div 3 = 105 \)[/tex] (since 315 is divisible by 3)
2. [tex]\( 105 \div 3 = 35 \)[/tex] (since 105 is divisible by 3)
3. [tex]\( 35 \div 5 = 7 \)[/tex] (since 35 is divisible by 5)
4. [tex]\( 7 \)[/tex] is a prime number.
So, the prime factorization of 315 is:
[tex]\[ 315 = 3 \times 3 \times 5 \times 7 \][/tex]
[tex]\[ = 3^2 \times 5 \times 7 \][/tex]
Step 2: Prime Factorization of 395
1. [tex]\( 395 \div 5 = 79 \)[/tex] (since 395 is divisible by 5)
2. [tex]\( 79 \)[/tex] is a prime number.
So, the prime factorization of 395 is:
[tex]\[ 395 = 5 \times 79 \][/tex]
Step 3: Product of Prime Factors
The prime factors of 315 are: [tex]\(3, 3, 5, 7\)[/tex]
The prime factors of 395 are: [tex]\(5, 79\)[/tex]
Now, we multiply all these prime factors together to find the product:
[tex]\[ 3 \times 3 \times 5 \times 7 \times 5 \times 79 \][/tex]
Let's organize the calculation step-by-step:
First, multiply the factors of 315:
[tex]\[ 3 \times 3 = 9 \][/tex]
[tex]\[ 9 \times 5 = 45 \][/tex]
[tex]\[ 45 \times 7 = 315 \][/tex] (We know 315 is correct as this is the original number)
Now, multiply 315 by the prime factors of 395:
[tex]\[ 315 \times 5 = 1575 \][/tex]
[tex]\[ 1575 \times 79 \][/tex]
To calculate [tex]\(1575 \times 79\)[/tex]:
[tex]\[ 1575 \times 79 = 1575 \times (70 + 9) \][/tex]
[tex]\[ = 1575 \times 70 + 1575 \times 9 \][/tex]
[tex]\[ = 110250 + 14175 \][/tex] (Breaking it down for easier multiplication)
[tex]\[ = 124425 \][/tex]
Thus, the product of the prime factors of 315 and 395 is:
[tex]\[ 124425 \][/tex]
So, the final answer is:
[tex]\[ 124425 \][/tex]
