Prove that for all integers m>0 , m^2+5m+4 is composite.

Q: (Proof by cases: (p₁ V P2 V (Pn →q)) Prove or disprove. V Pn) → q = (P₁ → q)^(p2 →q)^^...^ "The 4th…

A: Let x be any integer.  We need to prov        x4 does not leave remainder 2 when divided by 3.…

Q: find a Positive integern for which the equality holds %3D

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Q: Prove that 1+1=2

A: To prove: 1 + 1 = 2

Q: Prove that, all integers x and y, x^2-4y≠2  1a. Prove that all numbers of the form 1007, 10017,…

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Q: Let a, b, and c be integers such that a^2+b^2=c^2. Prove at least one of a and b is even.

A: It has been given to us that a, b, c are integers and a2 + b2 = c2.…

Q: Prove a^2 +4a+3 is odd if and only if a is even.

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Q: K = (perfect squares of integers} under addition? Is it closed or not

A: K is the perfect square of integerNow we find whether it is closed or not.

Q: Prove the claim that there are no integers j and k such that j^2 + 3k^2 = 8

A: Since both j and k are being squared so if we prove the case for positive integers, that will prove…

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A: We know, Hilbert's Grand Hotel is a thought experiment which gives counterintuitive property of…

Q: Prove that there are 1000 consecutive positive integers that are not perfect squares.

A: Here I give an example. That proves the hypothesis.

Q: Prove that the sum of the squares of two odd integers cannot be the square of an integer.

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Q: The set A consists of nine positive integers, none of which has a prime divisor larger than six.…

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Q: Find all integers x (if there exist any at all) such that x^2 + 5x + 6 is prime. Justify that your…

A: We have to find all integers x (if there exist any at all) such that x^2 + 5x + 6 is prime.

Q: Prove the following using a direct proof.  If a,b, and c are integers such that b is a multiple of…

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Q: prove that for all integers m>2,m^2+m-6 is composite

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Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: Prove that if x is an even integer, thenx^2−6x+ 3is odd

A: Any integer of the form 2m+1 is odd.

Q: Given an odd integer a, establish that a^(2) + (a+2)^(2) + (a+4)^(2) +1 is divisiible by 12 using…

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Q: prove the difference of the squares of two z integers is always divisible by 16 (i.e. z1^2 - z2^2 =…

A: We prove that (z_1)^2-(z_2)^2=16k for some integer k, then we say that divisible by 16.

Q: Prove that 4 does not divide n2 +2 for any integer n.

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Q: Prove that (-1+ iB)" + (-1- i3)" has either the velue 2t! or the value if a B any integer (positive,…

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Q: Suppose that a and b are integers, c b = 3 (mod 19). Find the integer c that a) с%3D 13а (mod 19).…

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Q: Prove that 3a^1 is even if and only if a is an odd integer.

A: In the given question we have to prove that Prove 3a^1 is even if and only if a is an odd integer.

Q: let M be the greatest integer less than 30 such that M!(M+1)!/2 is a perfect square. Let N be the…

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Q: theorem

A: We use Fermat's theorem here, If is prime and for any number a such that gcd(a,p)=1 then ap-1…

Q: Vn € Zt [obeb(n, n + 1) = 1]

A: l  let n,n+1 are two consecutive positive integers let  the greatest common divisor i.e…

Q: Prove: 3(12)^n + 8 is divisible by 11 for all positive integers

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Q: and

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