Prime Factorization - Examples, Exercises and Solutions

To determine all the factors of the number 7, we will examine which integers between 1 and 7 divide it exactly:

• Check 1: Since $\frac{7}{1} = 7$, 1 is a factor.
• Check 2: $\frac{7}{2} = 3.5$, which is not an integer, so 2 is not a factor.
• Check 3: $\frac{7}{3} \approx 2.333$, which is not an integer, so 3 is not a factor.
• Check 4: $\frac{7}{4} = 1.75$, which is not an integer, so 4 is not a factor.
• Check 5: $\frac{7}{5} = 1.4$, which is not an integer, so 5 is not a factor.
• Check 6: $\frac{7}{6} \approx 1.167$, which is not an integer, so 6 is not a factor.
• Check 7: Since $\frac{7}{7} = 1$, 7 is a factor.

Therefore, the factors of 7 are $1$ and $7$.

These results correspond to choice 1: $1, 7$.

To determine the factors of the number $6$, we will follow these steps:

• Step 1: Begin by checking each number starting from $1$ up to $6$ to see if it divides $6$ evenly.

• Step 2: Check $1$. Since $1 \times 6 = 6$, $1$ is a factor.

• Step 3: Check $2$. Since $2 \times 3 = 6$, $2$ is a factor.

• Step 4: Check $3$. Since $3 \times 2 = 6$, $3$ is a factor.

• Step 5: Check $4$. Since $6$ is not evenly divisible by $4$, $4$ is not a factor.

• Step 6: Check $5$. Since $6$ is not evenly divisible by $5$, $5$ is not a factor.

• Step 7: Finally, check $6$. Since $6 \times 1 = 6$, $6$ is a factor.

All possible whole number products (pairs) that result in $6$ are $1 \times 6$, $2 \times 3$, $3 \times 2$, and $6 \times 1$.

However, when checking for unique prime factors as a particular approach in factors identification, $6$ breaks down into prime factors of $2$ and $3$.

Therefore, the primary distinct prime factors of $6$ are $2$ and $3$.

This correlates with choice 3:

• Choice $3$: $2,3$, which matches our factors.

Thus, the answer is correctly represented as the distinct prime factors $2, 3$ in the context of the problem requirements.

To find all the factors of 9, we will determine the divisors of the number 9 by testing each integer from 1 up to 9.

• Step 1: Test if 1 is a factor of 9. Since $9 \div 1 = 9$, 1 is a factor.
• Step 2: Test if 2 is a factor of 9. Since $9 \div 2 = 4.5$ (not an integer), 2 is not a factor.
• Step 3: Test if 3 is a factor of 9. Since $9 \div 3 = 3$, 3 is a factor.
• Step 4: Test if 4 is a factor of 9. Since $9 \div 4 = 2.25$ (not an integer), 4 is not a factor.
• Step 5: Test if 5 is a factor of 9. Since $9 \div 5 = 1.8$ (not an integer), 5 is not a factor.
• Step 6: Test if 6 is a factor of 9. Since $9 \div 6 = 1.5$ (not an integer), 6 is not a factor.
• Step 7: Test if 7 is a factor of 9. Since $9 \div 7 \approx 1.2857$ (not an integer), 7 is not a factor.
• Step 8: Test if 8 is a factor of 9. Since $9 \div 8 = 1.125$ (not an integer), 8 is not a factor.
• Step 9: Test if 9 is a factor of 9. Since $9 \div 9 = 1$, 9 is a factor.

The factors of 9 are 1, 3, and 9.

However, the problem might specifically be asking for the prime factorization where the number 9 decomposes into $3 \times 3$.

Therefore, the correct answer which matches the provided choices is $3, 3$.

To find the factors of 8, we'll use prime factorization.

• Step 1: Begin with the smallest prime number, 2.
• Step 2: Divide 8 by 2, which gives 4. Since 4 is even, divide by 2 again.
• Step 3: Divide 4 by 2, which gives 2. Divide 2 by 2 one more time.
• Step 4: This yields 1, so we've fully factored the number.

Thus, the prime factorization of 8 is $2 \times 2 \times 2$.

The factors of the number 8 are $2, 2, 2$.

Therefore, the correct answer is choice 4: $2, 2, 2$.

To solve this problem, we'll follow these steps:

• Identify all the factors of the number 4.
• Determine which of these factors are prime numbers.
• Check the answer choices to find the one that corresponds to the prime factorization of 4.

Now, let's work through each step:
Step 1: To find the factors of 4, we consider pairs of numbers that multiply to 4, such as $1 \times 4$ and $2 \times 2$.
Step 2: Among these factors, identify the prime numbers. The number 2 is the only prime factor, and it needs to be listed twice since $2 \times 2 = 4$.
Step 3: Looking at the answer choices, the choice that corresponds to the prime factorization of 4 is $2, 2$.

Therefore, the correct answer is $2, 2$.