Here are two conjectures: Conjecture 1: For all integers a,b and c, if a|b and a|c, then a|bc. Conjecture 2: For all integers a,b and c, if a|c and b|c, then ab|c. Decide whether each conjecture is true or false and prove/disprove your assertions.

Q: Prove that for all integers m and n,if m and m + n are even,then n is even

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Q: Show that, for given integers a, b, c, if gcd(a, c) = gcd (b, c) = 1, then gcd (ab, c) = 1

A: Given a,b,c∈Z+ And gcda,c=1 and gcdb,c=1 We have to show that gcdab,c=1 Since, gcda,c=1 and gcdb,c=1…

Q: Question 3: Prove or disprove: Given integers a, b and c, if a|b and a|c, then a|(b-c).

A: Solution:Given       a | b  and a | cTo prove a| (b-c)

Q: If a, b, and c are integers such that a|b and c\b, then ac|b. а,

A: To check the given statement true or false lets take look a couple of example as follows : Take a=2…

Q: Let a and b be integers. Show that if every integer that divides a also divides b, then a | b. (a…

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Q: Prove that for all integers a, b, and c if a divides b - 1 and a divides c - 1 then a divides bc…

A: If a number x divides y then y is a multiple of x.

Q: what is (a'+b,a+b) Wher e a and b are relatively prime integers that are not both Zero1

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: Prove that if a > 2 and b are integers, then a † b or a { (b+1).

A: Given that a≥2 and b are integers. We will prove it by contradiction. Let a|b and a|(b+1) then there…

Q: Prove that for all integers a, b, and c if a divides b then for all naturla numbers n a^n divides…

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Q: Suppose x and y are any integers. Prove that if x and y are odd, then x + 5y is even

A: Suppose x and y are odd integers. To show x+5y2 is even. Note: Square of an odd number is odd.…

Q: For all integers x and y, if 6x+y is odd, then x and y are both odd. True False

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Q: Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

A: Let a, b and c be integers.

Q: Show that if a1,: a2., an are integers that are not all 0 and c is a positive integers, then…

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Q: Prove that, if a is an irrational number, then Va is also an irrational number.

A: In the given problem we have to prove that if x is an irrational number, then x3 is also an…

Q: -ん 2 kodd

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Q: Prove by using a direct proof that if n is an odd number, then n2 is also odd.

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Q: Prove: For all integers a and b, if a + b is odd then exactly one of the integers, a or b, is odd.

A: (1) We have to prove that a+b is odd then exactly one of the integer a or b is odd. Suppose that a…

Q: Supposed a and b are relatively prime integers and c is an integer such that a | c and b | c. Prove…

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Q: 3. Prove by contradiction, that if x and y are odd integers, then there is no integers z such that…

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Q: Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the remainder…

A: we have to Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the…

Q: Let a and b be two integers. Prove that if a+b is even then a-b is even.

A: Given that a and b are any two integers. We are trying to prove "if a+b is even then a-b is even."

Q: Suppose a and b are integers that divide the integer c. If a and b arerelatively prime, show that ab…

A: Since a/c   &  b/c  , we know there exists integers K1 & K2  ,with K1 a=K2 b=c--1Also…

Q: Prove or disprove. There are integers a, b, and c such that a divides bc, but a does not divide b…

A: We have to prove or disprove that if integers a, b, and c such that a divides bc,but a does not…

Q: Prove that 36n is not O(35n)

A: Let, 36n∈O35n

Q: Prove that for all integers a, b, and c, if a2 + b2 = c2, then at least one of a and b is even.

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Q: Suppose that k and n > k are fixed positive integers. Justify the identity k+1,

A: Given k and n≥k are fixed positive integers.

Q: 12. Prove that for all integers a and m, if a and m are the lengths of the sides of a right triangle…

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Q: Prove in a thoroughly written proof that for all integers a  and m, if a and m are the lengths of…

A: Recall the following. Pythagoras theorem: For any right triangle, the square of the hypotenuse is…

Q: Prove that if m and n are integers and mnis even, thenm is even or n is even.

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Q: If a, b, and c are integers such that b is a multiple of a and e is a multiple of b², then c is a…

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Q: Prove the statement by: a) contraposition b) contradiction

A: Given statement is, “For all integers m and n, if mn is even then m is even or n is even.

Q: Prove that the difference between two integers is odd if and only if exactly one of the integers is…

A: Suppose x be any integer. If x is odd then x=2m+1 for some integer m. If x is even then x=2m for…

Q: Prove that for every positive integer n the number (1 – /2)" is not ratioanal.

A: Given: The given number is, (1-2)n where n is the positive integer.   To prove: The number (1-2)n is…

Q: Which of the following statements is true: For all integers a, b and e, if (a + b)lc and then a | c…

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Q: Prove the following by contrapositive: If x is an even integer, then x^2 is even.

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Q: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even.

A: Given: Show that if n is a positive integer, then n is even if and only if 7n + 4 is even

Q: 4. Prove that if m and n are positive integers and x is a real number, then

A: We have to prove here that the expression here follows both commutative and associative property. If…

Q: Let a, b, and c be integers. Prove that if a divides b and a divides c, then a divides (b-c).

A: a, b and c are integers, a divides b and a divides c. Since a divides b, therefore b=ak,  k is any…

Q: 4. For natural numbers n and k, prove (C) - ()

A: We have to prove

Q: Prove that if m and n are integers and mn is even, then m is even or n is even.

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Q: Prove that, if z is an irrational number, then Va is also an irrational number.

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Q: Suppose x and y are any integers. Prove that if x and y are odd, then x + 5y? is odd.

A: We have to solve given problem:

Q: Theorem 2.8. For positive integers a and b gcd(a, b) lcm(a, b) = ab Froof. To begin, put d = gcd(a,…

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Q: Show that for all positive integers n and k, (~^)= (-1)^(" + k-1). k

A: The provided information is  (  k-n) Then we prove this (   k-n)=(-n)*(-n-1)*(-n-2)*.....(-n-k+1)k!…