a. Determine the prime-power factorizations of both 42 and 56 . b. Determine the set of all positive divisors (factors) of 42 . c. Determine the set of all positive divisors of 56 . d. Determine the greatest positive divisor of both 42 and 56 . a. The prime-power factorization of 42 is

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To determine the prime-power factorization of 42 and 56, we need to find the prime factors of each number and express them in the form of $$p^a \cdot q^b \cdot r^c$$, where $$p$$, $$q$$, and $$r$$ are prime numbers and $$a$$, $$b$$, and $$c$$ are their respective powers.

a. Prime-power factorization of 42:
The prime factorization of 42 is $$2 \cdot 3 \cdot 7$$.

b. Set of all positive divisors of 42:
The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.

c. Set of all positive divisors of 56:
The positive divisors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56.

d. Greatest positive divisor of both 42 and 56:
The greatest positive divisor of both 42 and 56 is 14.

Therefore, the prime-power factorizations are:
- 42: $$2^1 \cdot 3^1 \cdot 7^1$$
- 56: $$2^3 \cdot 7^1$$

The set of all positive divisors for 42 is: 1, 2, 3, 6, 7, 14, 21, 42
The set of all positive divisors for 56 is: 1, 2, 4, 7, 8, 14, 28, 56
The greatest positive divisor of both 42 and 56 is 14.
a. Prime-power factorization of 42: $$2^1 \cdot 3^1 \cdot 7^1$$ b. Positive divisors of 42: 1, 2, 3, 6, 7, 14, 21, 42 c. Positive divisors of 56: 1, 2, 4, 7, 8, 14, 28, 56 d. Greatest positive divisor of both 42 and 56: 14

a. Determine the prime-power factorizations of both 42 and 56 . b. Determine the set of all positive divisors (factors) of 42 . c. Determine the set of all positive divisors of 56 . d. Determine the greatest positive divisor of both 42 and 56 . a. The prime-power factorization of 42 is

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