24. Determine whether the following assertions are true or false. If true, prove the result, and if false, give a counterexample. (1) If (a, b) = (a, c) then [a, b] = [a, c]. (2) If (a, b) = (a, c) then (a^2, b^2) = (a^2, c^2). (3) If (a, b) = (a, c) then (a, b) = (a, b, c). (4) If p is a prime and p | a and p | (a^2 + b^2) then p | b. (5) If p is a prime and p | a^7 then p | a.

          24. Determine whether the following assertions are true or false. If true, prove the result, and if false, give a counterexample.
(1) If (a, b) = (a, c) then [a, b] = [a, c].
(2) If (a, b) = (a, c) then (a^2, b^2) = (a^2, c^2).
(3) If (a, b) = (a, c) then (a, b) = (a, b, c).
(4) If p is a prime and p | a and p | (a^2 + b^2) then p | b.
(5) If p is a prime and p | a^7 then p | a.

24. Determine whether the following assertions are true or false. If true, prove the result, and if false, give a counterexample.
(1) If (a, b) = (a, c) then [a, b] = [a, c].
(2) If (a, b) = (a, c) then (a^2, b^2) = (a^2, c^2).
(3) If (a, b) = (a, c) then (a, b) = (a, b, c).
(4) If p is a prime and p | a and p | (a^2 + b^2) then p | b.
(5) If p is a prime and p | a^7 then p | a.

Added by Juan Manuel C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Hoan Nguyen

03/10/2023

Step 1

Proof: Since $(a, b) = (a, c)$, there exists a common divisor $d$ such that $d|a$ and $d|b$ and $d|c$. Now, we know that $[a,b] = \frac{ab}{(a,b)}$ and $[a,c] = \frac{ac}{(a,c)}$. Since $(a, b) = (a, c)$, we can write $[a,b] = \frac{ab}{d}$ and $[a,c] = Show more…

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24. Determine whether the following assertions are true or false. If true, prove the result, and if false, give a counterexample. 1. If (a, b) = (a, c) then [a,b] = [a,c] 2. If (a, b) = (a, c) then (a^2, b^2) = (a^2, c^2) 3. If (a, b) = (a, c) then (a, b) = (a, b, c) 4. If p is a prime and p | a and p | (a^2 + b^2) then p | b 5. If p is a prime and p | a^7 then p | a