Prove the following statements. If a statement is not true, find a counterexample. 1. The difference of any two even integers is even. 2. For every integer m, if m > 2, then m²-4 is composite. 3. For all integers a, b, c, and d, if a c and b|d, then ab | cd.

Q: 13. Recall Z = {..,-2,–1,0,1,2,..} (the integers). Let Z* = {1,2,3,.} be the positive integers. Let…

A: Given: ℤ=…, -2, -1, 0, 1, 2, … and ℤ+=1, 2, 3, …

Q: 74. Prove that for any real numbers a and b, |b − a|=|a − b|.

A: Given; a and b are any real numbers Consider, L.H.S=b-aWe can write it as,=b+-a=-1-b+a=-1a-b…

Q: If m and n are integers and m x n = 0, then what is true for the values of m and n?

A: If m and n are integers and m×n=0

Q: For any real number a, a ≠ 0, show that a+4 a+7 a+5 a +8 = 0, a+6 а+9 a+1 a+2 a+3 and 4a7a 5a 8a =…

A: We are required to prove: a+1a+4a+7a+2a+5a+8a+3a+6a+9=0a4a7a2a5a8a3a6a9a=0aa4a7a2a5a8a3a6a9=0

Q: Let a, b e R. Prove that if 15a + 206 = 49, then a and b are not both integers.

A: we have 15a+20b =49 and a and b both are in R  we prove that a and b are not both integers.

Q: Prove the statement if it is true, and give a counterexample if it is not true. a. For every real…

A: Both the subparts are solved with explanation.

Q: . Let a, b € Z so that 4a + 5b is a multiple of 13. Prove that 13 divides 9a + 8b.

A: We have to show that 13 divides 9a+8b To prove this we using 4a+5b divisible by 13

Q: If a = 2a - a² and b =1-(1-a), show that a = b for all natural numbers n. Assume that a, is a fixed…

A:

Q: Let a, b e R. Prove that if 15a + 206 = 49, then a and b are not both integers.

A:

Q: 3. Let x be an integer. Use the previous result to prove the following corollaries. (a) If x divides…

A: Divisibility: If a and b are integers, we say that a divides b, and write a|b, if there is an…

Q: If a, b, and c are consecutive positive integers such that a < b < cand abc is NOT a multiple of 4,…

A: Given: a, b and c are consecutive positive integers. a&lt;b&lt;c  ……1   and, abc is not a multiple…

Q: Determine whether the statement is true or false. Prove the statement directly from the definitions…

A: a. For all integers a,b, and c, if a | bc, then a | b or a | c.To demonstrate this assertion, we'll…

Q: Number 4 only. Thanks!

A:

Q: 1. Let a, b € Z so that 2a + 3b is a multiple of 11. Prove that 11 divides a - 46.

A:

Q: Disprove the following statements. (a) If n and m are natural numbers, then nm is composite. (b) For…

A: I have given an answer in step 2.

Q: 6. Let x and y be integer. Prove the following x + y is odd if and only if x² + y² is odd.

A:

Q: Let a and b be integers. Prove that GCD(a,b) divides GCD(a+b, a−b). Give an example when GCD(a,b)…

A:

Q: Determine whether the following statements are true or false. Justify your answer with a proof or a…

A:

Q: 2. Prove the statement: The sum of the squares of any two consecutive integers is odd.

A: I have assumed that n be any integer. Then i have found the sum of square of two consecutive…

Q: 3. Let a be a primitive 10th root of unity. Which of the following is false? A. a5 = 1 B. a³ = a¹3…

A:

Q: Using Discrete Mathematics and language. Determine whether the statement is true or false. Prove the…

A: The given statement, " For all integers  and , if  then  and .The aim is to check whether the…

Q: Let a be a real number. For 1 A = 2 - fınd A² . a -2 – 2a 4 + 4a 2a? – 4 -2 | - None of the others…

A: First option is correct. See the attachment.

Q: Decide whether R \ Q is countable or uncountable, and prove your answer. You can use what you know…

A: To prove that R \ Q is countable or uncountable.

Q: DIVISION BYt (:) - {}> 81. Show that In %3D

A:

Q: If the product of two integers is positive, their signs are . b. If the product of two integers is…

A: To solve the following problem

Q: Let m and n be integers. Recall that we say an integer a is even if a=2b for some integer b. We say…

A:

Q: 1. Let a and b be integers. Prove that GCD(a,b) divides GCD(a+b, a−b). Give an example when GCD(a,b)…

A: Let, GCD(a, b) = d1 =&gt; d1 | a and d1 | b=&gt; d1 | (a + b) and d1 | (a - b) =&gt; d1 is a common…

Q: Suppose a > 0, (a, 42) = 6 and [a, 42] = 420. Find а.

A:

Q: Let a, b e Z so that 2a + 36 is a multiple of 11. Prove that 11 divides a - 4b.

A:

Q: 2. Write as the sum/difference of logs. Express powers as factors. (r+2)71 In LyVz-5

A: Givenlnx+27yz-5To wrtite the sum/difference of log Formula:ln ab=ln a -ln bln ab=ln a + ln bln ak=k…

Q: Prove that for integers a and b, (a -3)b2 is even if and only if a is odd or b is even.

A: We need to prove that for integers a and b, (a -3)b^2 is even if and only if a is odd or b is even.