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

Q: Prove that for every integer n≥0, the number 42n+1+3n+2 is a multiple of 13.

A: Let us prove the given statement by mathematical induction. Case 1: Let n=0

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

A: -11+-12+-13+---+-1n=-1n-12. We prove this statement by using the Principle of Mathematical…

Q: Prove: For integers m, n, 12mn – 9 is an odd integer.

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

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

Q: Prove that if n is a positive integer, then n-1 is odd if and only if 5n + 13 is odd.

A: Given that n is any positive integer. Now, to prove that n-1 is odd iff 5n+13 is odd

Q: Prove or disprove: For every positive integer n, 22n – 1 is divisible by 3. -

A: I have proved the result using mathematical induction

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: Use indirect proof to show that, "If n is an integer and 3n+2 is even, then n is even".

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Q: For every positive integer n, prove that 7n – 3n is divisible by 4.

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Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169.

A: Topic - principle of mathematical induction

Q: Prove that if n is a positive integer, then 133 divides 11"+1 + 122n-1

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Q: Prove that the statements here are true for every positive integer n. 4 + 42 + 43 + · · · +…

A: To prove: For every positive integer, 43+44+......+4n=4(4n-16)3

Q: Prove that if n is an integer and 5n + 2 is even, then n is even using a proof by contraposition. a…

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

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

Q: Prove that n²– 3n + 1 is odd for every integer n.

A: To prove that n^2-3n+1 is odd for every integer n.

Q: Prove by using proof by contradiction: If (3n)^2 is even, then n is even.

A: Any odd integer multiply by an odd integer is odd. Any even integer multiply by an even integer is…

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

A: We know that the sum of a geometric series of the form: a+ar+ar2+ar3+...........+arn-1 with the…

Q: Prove that for all natural numbers n, (n^3/3) + (n^5/5) + (7n/15) is an integer

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Q: Prove that 8 | (54" +7) for every positive integer n.

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Q: Use contradiction to prove:  if 3n + 2 is an odd integer, then n is odd.

A: consider 3n+2 as odd integer  and let  n is even  therefore, n=2k for some integer k therefore,…

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

A: Given,  2+7+12+...+(5n-3)=12n(5n-1)

Q: Prove that the statements here are true for every positive integer n. 4 + 42 + 43 + · · · +…

A: To side equations are given, 4 + 42 + 43 + · · · + 4n=4(4n-1)/3We need to find out if above equation…

Q: Prove that for all integers m and n, if m is odd and the remainder is 2 when n is divided by 3, then…

A: Given, m is odd  When n is divided by 3, then the remainder is 2  We have to prove when  3m+2n  is…

Q: Prove that if n is an integer and 5n + 6 is even, then n is even.

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

A: we have to prove that the statement is true for every positive integer n.…

Q: Prove that 12 +32 +52 + ...+ (2n + 1)? = (n + 1) (2n + 1)(2n + 3)/3 whenever n is a nonnegative…

A: To prove: That 12+32+52+...+2n+12=n+12n+12n+33 where, n is non negative number. Calculation: Use…

Q: Prove that the statement is true for every positive integer n.3 is a factor of n3 + 2n

A: Consider the given statement, "3 is a factor of  n3+2n" for all positive integer n Suppose Pn…

Q: Prove that if n is an integer and n² is an even integer, then n is an even integer

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Q: Prove by using proof by contrapositive:   Prove that if 3n + 2 is odd, then n is odd.

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Q: Prove that if n is an integer and 3n + 2 is even, then n is even using a) a proof by contraposition

A: Given , n is an integer .To prove → if 3n+2 is even , then n is even using a proof by composition…

Q: Prove that for any integer n, n' -n is even.

A: CASE I When n is even Let n=2m,m∈N⇒n3−n=2m3−2m⇒2m4m2−1=2η,where η=m4m2−1Thus,n3−n is even when n is…

Q: Prove that for all integers n, if n² is even then n is even.

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Q: prove that if a and b are both odd integers, then ab+1 is an even integer

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Q: Prove that for every positive integer n, there are n consecutive composite integers.

A: This question is related to Descrete Mathematics

Q: . Let n be an integer. Prove that n is odd if and only if 3n? – 7 is even.

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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 3n^2 + 1 is even if and only if n is an odd integer.

A: We need to prove that 3n2+1 is even if and only if n is an odd integer.

Q: Prove or disprove that for every positive integer n, if n is even then n^{2}+1 is prime.

A: False

Q: there are no integers x and y such that x is a prime greater than 5 and x=6y+3

A: Given query is to check the statement whether it's possible or not.

Q: Prove that 3, 5, and 7 are the only three consecutive odd integersthat are prime.

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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: Prove that the statements here are true for every positive integer n. 12 + 22 + 32 + · · · + n2=n(n…

A: Proved that for n≥1 fn=12+22+32+⋯⋯+n2=nn+12n+16 For n=1 LHS=12=1 RHS=nn+12n+16=11+12+16=66=1…

Q: Prove that the statements here are true for every positive integer n. 1 + 3 + 5 + · · · + ( 2n - 1)…

A: Show that for every positive integer n    1 + 3 + 5 + · · · + ( 2n - 1) = n2

Q: Show that for every integer n ≥ 8, there exist integers a ≥ 0 and b ≥ 0 such that n = 3a + 5b.

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Q: Prove that if n is an integer, then 2n2 + n + 1 is not divisible by

A: The given problem is to prove the following statement using the possibile cases. We need to find the…

Q: Prove that n2 − 3n + 1 is odd for every integer n.

A: Prove the result using mathematical induction. P(n):n2-3n+1. Check whether the result is true for n…

Q: Prove that for any integer n, if n³ is odd then n is odd.

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

A: a>1, then an>1

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!…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.