Prove: if a and b are integers such that a is a factor of b,  then for any integer c, a is a factor of b*c.

Q: Prove that if a and b are integers with a 0 and where + b - a = 0, x is a positive integer such that…

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Q: Prove that if a is an odd integer, then 5a² – 3a + 5 is odd.

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Q: 10) Let a and b be positive integers. Prove that (a, b) = a if and only if a divides b.

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Q: Prove that for any positive integer n, 6(7") – 2(3") is divisible by 4.

A: We use binomial expansion formula for an-bn to solve the above problem.

Q: 23. Prove that for any integer a, 9/ (a² – 3). -

A: Prove that for any integer a, 9 ⫮ (a2 – 3).

Q: Prove that, nonetheless, for all integers a, if 2 | a and 3 | a, then 6 | a

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Q: = Let a, b, and d be positive integers. Prove that if (a, b) : divides a, then (d, b) = 1.

A: Definition of gcd (a, b) =1 is a and b does not have common prime divisor. Using this we can easily…

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: 2. Prove: If a, b, c, and d are integers such that a + b =c+d, then a A=

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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: Let a and b be two integers such that b | a. Let p be a prime such that p|a and p{b. Prove that p|…

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Q: |Let a, b, and c be integers. Prove that if gcd(a, b) | bc, then a | 1 and gcd(a, c) 1 and a (a) C.…

A: To prove the required divisibility properties under the given conditions on the greatest common…

Q: Prove that the statement is true for every positive integer n.

A: Given that, 2 is a factor of n2+n   We have to prove this statement is true for every positive…

Q: Prove that for all integers a,b which are not both 0, and for all integers d, d|gcd(a,b) if and only…

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Q: Let a and b be integers with b # 0. Let q and r be the quotient and remainder yielded by the…

A: Given: a and b are integers with b≠0 and q, r are quotient and remainder yielded by division theorem…

Q: Prove that if for some integers a, b, c we have that a + b3 + c* is divisible by 9, then at least…

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Q: Use proof by division into cases to prove the following statement: For all nonzero integers a, Hal –…

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Q: Prove that for all the integers a and n, if a | n, then a^2 | n^2

A: Given that a | n. This means n=ak, where k=integer. Then we square both sides: n2=(ak)2 or n2=a2k2

Q: Prove that for any four distinct and positive integers, there are two of them, say a and b, such…

A: Prove that for any four distinct and positive integers, there are two of them, say a and b, such…

Q: Prove that for any integer a , one of the integers a,a+2,a+4 is divisible by 3

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Q: Prove: that if a is an odd integer, then 4 is a factor of a^2 -1

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Q: Prove that if a, b, and e are integers such that a l b and a l e, then a l (b + e).

A: Given: a, b, and e are integers such that a l b and a l e then  To prove: a l (b + e).

Q: Prove that if a and b are integers such that a|b, then either a = b or a = -b

A: Take a = 2 ,b = 4 . We have a divides b but neither a = b , nor a = -b.

Q: S Prove by mathematical induction that x 1 is a factor of x" 1 for all integers n > 1.

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Q: Prove that the statement is true for every positive integer n.

A: We have, Pn=a2n-1+b2n-1 Let us consider the statement Pn is true for ∀n∈ℕ. Therefore,…

Q: Consider these four statements: A: For all integers a, b, and c, if a | b and a | c then a|(2b –…

A: The objective is to find the correct statement.

Q: a. Prove that for every integer a, if a³ is even then a is even. b. Prove that 2 is irrational.

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Q: Prove that if c is a natural number then for all a and b we have : gcd(ac, bc) = c gcd(a, b)

A: We have to prove that if c is a natural number then for all a and b gcd(ac, bc)=c gcd(a, b) .

Q: If a and b are nonzero integers and m is the least common multiple of a and b, prove that (a) ∩ (b)…

A: Consider the given statement. (a) ∩ (b) = (m) Consider the left side of the equation. As, m is the…

Q: prove that Every positive integer n can be written as n =( 2^a)(5^b)c, where c is not divisible by 2…

A: Fundamental Theorem of Arithmetic: Every positive integer (except n=1) can be uniquely expressed as…

Q: For any integers, a, b, c, show that if a|b and a|c, then a|(k1b+k2c) for any k1, k2∈Z.

A: Division Algorithm: Let x and y be two integers with y>0.  Then there exist unique integer q and…

Q: Prove: If a is a positive even integer and b is an odd integer, then (a,b)=(a/2, b).

A: Let d be gcd of a and b. Now, b is odd so d cannot be even or is odd.  Now, since d is odd and…

Q: 11. Prove that for each integer a, if a? - 1 is even, then 4 divides a?

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Q: Prove that, for all integers K and L, there is at least one pair of integers (a, b) for which K2 +…

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Q: Prove: For all integers a, b, and c, if alb and alc then a (2b+3c)

A: prove is given below ....

Q: prove For all integers a, b, c, and d, if a|c and b|d then ab|cd.

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Q: (a) Let a,b, and c be integers such that a | b and a | c, and let x and y be arbitrary integers.…

A: According to the guidelines I am answering first question only. a) a, b and c be integers such that…

Q: Prove that there do not exist positive integers a and n such that a? + 3 = 3".

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Q: 2. Show that if a and b are positive integers and alb, then a < b. 3. Show that if a and b are…

A: Basic facts about division will be used in solving this problem.

Q: Let suppose integers are randomly selected. Prove that the difference of two of them is divisible by…

A: Solution

Q: rove that if x and y are odd positive integers ,then x+y is even but not divisible by 4

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Q: Prove or disprove: For all integers a and b, if a +b is odd, then a is even and b is odd.

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Q: Prove that the statement is true for every positive integer n.

A: a&gt;1, then an&gt;1

Q: Prove or disprove:  For all integers a and b, if a+b is even, then a is even and b is even.

A: We disprove this statement by taking a counter example.