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