Question

Prove the following statements using either direct or contrapositive proof. 18. If a,b∈Z,then (a+b)^3 ≡ a^3+b^3...

Prove the following statements using either direct or contrapositive proof.

18. If a,b∈Z,then (a+b)^3 ≡ a^3+b^3 (mod 3).

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

Here I'm using divided definition.We know 3|3 then 3 |(3* any nonzero integer).Also a|b and a|c then a|(b+c).Here I'm using (a+b)^3=a^3+3a^2b+3ab^2+b^3.also.proof is below thank you.

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Prove the folllwing case and use the Contrapositive approach to the proof. If n^2 + 2...
Prove the folllwing case and use the Contrapositive approach to the proof. If n^2 + 2 is not divisible by 3, then n is divisible by 3. First state the contrapositive before you begin the proof.
Use the method of direct proof to prove the following statements. 26. Every odd integer is...
Use the method of direct proof to prove the following statements. 26. Every odd integer is a difference of two squares. (Example 7 = 4 2 −3 2 , etc.) 20. If a is an integer and a^ 2 | a, then a ∈ { −1,0,1 } 5. Suppose x, y ∈ Z. If x is even, then x y is even.
Prove the following using the specified technique: (a) Prove by contrapositive that for any two real...
Prove the following using the specified technique: (a) Prove by contrapositive that for any two real numbers,x and y,if x is rational and y is irrational then x+y is also irrational. (b) Prove by contradiction that for any positive two real numbers,x and y,if x·y≥100 then either x≥10 or y≥10. Please write nicely or type.
3. Prove by contrapositive: Let n ∈ N. If n^3−5n−10>0,then n ≥ 3. 4. Prove: Letx∈Z....
3. Prove by contrapositive: Let n ∈ N. If n^3−5n−10>0,then n ≥ 3. 4. Prove: Letx∈Z. Then5x−11 is even if and only if x is odd. 4. Prove: Letx∈Z. Then 5x−11 is even if and only if x is odd.
Prove the statements (a) and (b) using a set element proof and using only the definitions...
Prove the statements (a) and (b) using a set element proof and using only the definitions of the set operations (set equality, subset, intersection, union, complement): (a) Suppose that A ⊆ B. Then for every set C, C\B ⊆ C\A. (b) For all sets A and B, it holds that A′ ∩(A∪B) = A′ ∩B. (c) Now prove the statement from part (b)
Write the contrapositive statements to each of the following. Then prove each of them by proving...
Write the contrapositive statements to each of the following. Then prove each of them by proving their respective contrapositives. a. If x and y are two integers whose product is even, then at least one of the two must be even. b. If x and y are two integers whose product is odd, then both must be odd.
Write the contrapositive statements to each of the following.  Then prove each of them by proving their respective contrapositives. ...
Write the contrapositive statements to each of the following.  Then prove each of them by proving their respective contrapositives.  In both statements assume x and y are integers. a. If  the product xy is even, then at least one of the two must be even. b. If the product xy  is odd, then both x and y must be odd. 3. Write the converse the following statement.  Then prove or disprove that converse depending on whether it is true or not.  Assume x...
Prove or disprove the following statements. a) ∀a, b ∈ N, if ∃x, y ∈ Z...
Prove or disprove the following statements. a) ∀a, b ∈ N, if ∃x, y ∈ Z and ∃k ∈ N such that ax + by = k, then gcd(a, b) = k b) ∀a, b ∈ Z, if 3 | (a 2 + b 2 ), then 3 | a and 3 | b.
Using either proof by contraposition or proof by contradiction, show that: if n2 + n is...
Using either proof by contraposition or proof by contradiction, show that: if n2 + n is irrational, then n is irrational. Using the definitions of odd and even show that the following 4 statements are equivalent: n2 is odd 1 − n is even n3 is odd n + 1 is even
1. ∀n ∈ Z, prove that if ∃a, b ∈ Z such that a 2 +...
1. ∀n ∈ Z, prove that if ∃a, b ∈ Z such that a 2 + b 2 = n, then n 6≡ 3 (mod 4).
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT