Prove that 3" + 1 is divisible by 4 whenever n is an odd positive integer.

Q: Prove that any positive integer greater than or equal to 8 can be written as a sum of 3's and 5's.…

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Q: 2. Prove that 11" – 7" is divisible by 4 for any positive integer п.

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Q: Prove that for all natural number n, 52n-1 is divisible by 8.

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Q: If n2 4 is an integer,prove by induction that n<nl.

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Q: Prove the statement. "For any integer n, nº +4 is not divisible by 8." by Contradiction.

A: Given: n be any integer. To prove: n3+4 is not divisible by 8.

Q: Prove by induction that n3 -n is divisible by 6 for all positive integers

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Q: 4) Prove that the product of any three consecutive integers is divisible by 6.

A: We have to show that product of any three consecutive integers is divisible by 6

Q: Use a direct proof to show that the product of two odd integers is odd.

A: Introduction: A direct proof is a series of statements, the final of which is the conclusion to be…

Q: Show by contraposition that for all integers n^2, if  is odd, then n + 1 is even.

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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: 2. Prove that the product of any two odd integers is an odd integer.

A: Let x and y be two odd numbers. If a and b are odd, then a and b can be written as: x=2m+1 , y=2n+1…

Q: Let n be an integer. Prove that if 2n +4 is divisible by 4, then n is even.

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Q: of the first 150 positive integers?

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Q: Prove that 5n-1 is divisible by 4 to any positive integer n

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Q: How can I prove by induction that 2^n + 1 is divisible by three for all odd integers

A: To apply mathematical induction to show that 2^n+1 is divisible by 3 if n is odd

Q: Prove: If x is a positive prime integer, then x+7 is a composite integer. (i.e., not prime)

A: See solution below

Q: Prove that for any odd positive integer, n, the sum of nconsecutive positive integers is divisible…

A: To prove for any odd positive integer, n, the sum of nconsecutive positive integers is divisible by…

Q: 75. Prove that the sum of an even integer and an odd integeris odd.

A: sum of an even integer and an odd integer is odd.

Q: rove that the difference of any two odd integers is even.

A: Even numbers are of the form 2k. Odd numbers are of the form 2k+1, where k is an integer.  Let a and…

Q: Prove that 2^10 + 5^12 is a composite number.

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Q: If n is a positive integer prove that 3^(2n) - 26n -1 is divisible by 676

A: if n is a postive integer prove that 3^(2n) - 26n -1 is divisible  by 676

Q: Prove that an odd number multiplied by an odd number is odd.

A: The objective is to prove that an odd number multiplied by an odd number is odd.

Q: Prove that n2-n is divisible by 2 for every positive integer n

A: We have to prove that,  n2-n is divisible by 2 for every positive integer n. We will use…

Q: Prove by Mathematical Induction that 10" - 1 is divisible by 9 for all positive integers n.

A: In proving some statements for all positive integers, we follow the mathematical induction. The…

Q: Suppose thatk and n 2k are fixed positive integer. Justify the identity k+1 jk

A: to prove ∑j=knjk=n+1k+1above equation can be written as S2(j,k)=S2(j-1,k-1)+k.S2(j-1,k)count the…

Q: 5. Prove that the integer 3". 22" – 1 is divisible by 11 for all n > 0.

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Q: イの Prove by mathematical induction that 3" + 7 is divisible by 8 for all integers n > 0

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Q: Prove that n- + 1> 2" when n is a positive integer with 1 <n<4.

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Q: 7. Show that for every natural number n2 1, the integer ) + () +*(;) + - +) +22 +... +2" is…

A: Using binomial theorem:  n0+2n1+22n2....2nnn=n01n·20+n11n-1·21+n21n-2·22....nn1n-n·2n…

Q: Prove that the product of an even number and an odd number is even.

A: Here, objective is to prove that the product of an even number &amp; an odd number is even

Q: 3. Use induction to prove that 15"+2 + 162n+1 is divisible by 241 for every nonnegative integer п.

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Q: prove that the one's digit of any integer,n, is the remainder when one divides n by 10.

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Q: Show that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer.

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Q: Show that there are infinitely many integers n such that 4(n^2)+1 is simultaneously divisible by 13…

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Q: 7. Without using any theorems from the book, prove the following statement: Let n be an integer. If…

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Q: 6. Prove that 3" < n! whenever n is a positive integer greater than 6.

A: We need to prove that ,  3n &lt; n!  for every positive integer n greater than 6. We will use method…

Q: Use a direct proof to show that if n² is an odd integer, then n* is also an odd integer.

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Q: Prove that 2 - 2 · 7 + 2 · 72 – ... + 2(-7)" = (1 – (-7)n+1)/4 whenever n is a nonnegative integer.

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Q: Prove that 2^(27) - 1 is not a prime number.

A: A prime number is a natural number greater than 1 that is not a product of two smaller natural…

Q: Prove that any integer greater than or equal to 48 can be represented as the sum of nonnegative…

A: Principle of mathematical induction: For each and every natural number n, mathematical induction is…

Q: Prove that n2-n is divisible by 2 for every positive integer n.

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Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

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Q: Prove that for any odd positive integer, n, the sum of n consecutive positive integers is divisible…

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Q: Prove 8^ +l is composic for any poSitive integer n.

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Q: Assuming every integer is either even or odd, How would I show that for any integer, the difference…

A: Assume that every integer is either even or odd. Let the integer considered say x is even. Then by…

Q: Use mathematical induction to prove 8^n − 1 is a multiple of 7 whenever n is a positive integer.

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Q: Prove by induction that (2) = n22"+1 – n2"+2 +3.27+1 - 6 whenever n is a positive integer

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Q: Prove that any positive integer N is divisible by 11 if and only if the difference between the sum…

A: Consider the provided question, we need to prove that, Any positive integer N is divisible by 11 if…