do you understand?
1. essential question how can you write the prime factorization and find the greatest common factor and the least common multiple of two numbers?
2. what are two different ways in which you can use prime factorization to find the prime factors of a number?
3. generalize why is the gcf of two prime numbers always 1?
4. construct arguments in example 4, grant finds applesauce that comes in packages of 8, but now he finds juice bottles in only packages of 3. will the lcm change? explain.
5. critique reasoning sarah says that you can find the lcm of any two whole numbers by multiplying them together. provide a counterexample to show that sarah’s statement is incorrect.

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Accepted Answer

### Question 2
# Brief Explanations:
One way is the factor tree method: start with the number, split it into a product of two factors, then split those factors until all are prime. Another way is the division method: divide the number by the smallest prime (starting from 2) that divides it, then divide the quotient by the smallest prime, and repeat until the quotient is 1; the divisors are the prime factors.
# Answer:
- Factor Tree Method: Draw a factor tree by repeatedly breaking the number into factor pairs until all factors are prime. For example, for 12, we can write $12 = 2\times6$, then $6 = 2\times3$, so prime factors are $2, 2, 3$.
 - Division Method: Divide the number by the smallest prime divisors successively. For 12, divide by 2 ($12\div2 = 6$), then divide 6 by 2 ($6\div2 = 3$), then divide 3 by 3 ($3\div3 = 1$). The prime factors are the divisors used: $2, 2, 3$.

### Question 3
# Brief Explanations:
A prime number has only 1 and itself as factors. For two different prime numbers, the only common factor they share is 1, since their only factors are 1 and themselves (and the two primes are different, so themselves are not common). So the greatest common factor (GCF) is 1.
# Answer:
A prime number $p$ has factors $1$ and $p$, and a prime number $q$ (where $p
eq q$) has factors $1$ and $q$. The common factors of $p$ and $q$ are only $1$ (since $p
eq q$, so $p$ is not a factor of $q$ and vice versa). Thus, $\text{GCF}(p, q)=1$.

### Question 4
# Brief Explanations:
Assume originally the packages were, say, 8 (applesauce) and another number (let's assume the original juice packages were, for example, 4, but now it's 3). The LCM of two numbers depends on their prime factorizations. The prime factorization of 8 is $2^3$, and of 3 is $3^1$ (since 3 is prime). If the original juice package number had a different prime factorization (e.g., 4 is $2^2$), the LCM would change. The LCM is the product of the highest powers of all primes present. So changing one number (from, say, 4 to 3) changes the prime factors involved (adding 3 and removing the extra 2s), so the LCM will change. For example, if originally we had 8 and 4: LCM(8, 4) is 8 (since $8 = 2^3$, $4 = 2^2$, highest power of 2 is $2^3 = 8$). Now with 8 and 3: LCM(8, 3) is $2^3\times3 = 24$, which is different from 8.
# Answer:
Yes, the LCM will change. Let's assume originally the juice packages were, for example, 4 (and applesauce is 8). $\text{LCM}(8, 4)=8$ (since $8 = 2^3$, $4 = 2^2$, LCM takes the highest power of 2, $2^3 = 8$). Now, with juice packages of 3 (a prime number, $3 = 3^1$) and applesauce of 8 ($2^3$), the LCM is calculated as the product of the highest powers of all primes: $2^3\times3^1 = 24$. Since $24
eq8$, the LCM changes because the prime factorizations of the two numbers (the package counts) have changed, and LCM depends on the prime factors and their highest powers.

### Question 5
# Brief Explanations:
To find a counterexample, we need two whole numbers where multiplying them is not equal to their LCM. Take 2 and 4. Multiply them: $2\times4 = 8$. Now find the LCM of 2 and 4. The prime factorization of 2 is $2^1$, and of 4 is $2^2$. The LCM is the highest power of 2, which is $2^2 = 4$. Since $8
eq4$, this shows Sarah's statement is incorrect.
# Answer:
Let's take the numbers 2 and 4.
- Multiply them: $2\times4 = 8$.
- Find the LCM of 2 and 4: Prime factorization of $2 = 2^1$ and $4 = 2^2$. The LCM is the highest power of the prime factor (2) present, so $\text{LCM}(2, 4)=2^2 = 4$.
Since $8
eq4$, multiplying 2 and 4 does not give their LCM, so Sarah's statement is incorrect.

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