4. Use induction to prove that 64 divides 32n+2 + 56n + 55 for every positive integer n.

Q: Determine whether there are infinitely many primes that have a remainder of 12 when divided by 25.…

A: Given question is Determine whether there are infinitely many primes that have a remainder of 12…

Q: For any integer z, if 3 divides z2, then 3 also divides z.

A:

Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: 3n+2 and 5n+3 are relatively prime for every integer n.

Q: Prove that if n is an integer and 7n+2 is even, then n is even. State which method you are using for…

A: Let  n be any integer.  We have given that , 7n + 2 is even. We need to prove that ,  n is even. We…

Q: 32. Prove that 3 divides n³ +2n whenever n is a positive integer.

A:

Q: Prove by the method of Mathematical Induction that, 3 divides n + 2n for any positive integer n.

A:

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

A:

Q: Show that an integer N is divisible by 9 if and only if the sum of its digits is divisible by 9

A: Let a=anan-1...a3a2a1 be an integer and S be the sum of the digits. Consider a,…

Q: exp(ix())= 4k+1 2

A:

Q: 3. Let a and b be positive integers. Express the positive greatest common divisor of a and b in…

A: The positive greatest common divisor (gcd) of two positive integers, a and b, can be expressed in…

Q: Show that for any integer n , n2 – 2 is not divisible by 3.

A:

Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

A: The objective is to prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

Q: 4. Show that for all positive integers n, n! + 1 has a prime divisor greater than n.

A: Let n be an integer such that n>1. All prime divisors of n!+1 must be bigger than n. This is…

Q: divisible by 3 for

A:

Q: 11n+1 +122n-1 is divisible by 133 for all natural numbers

A:

Q: Disprove  For any integer n, n2 + 5 is not divisible by 5.

A:

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

A: Given, n² is an odd integer. So, n² = 2k+1, where k is an integer. Now, n⁴ = (n²)² = (2k+1)² = 4k² +…

Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: We have to prove that and  are relatively prime for every integer .We know thatThe greatest common…

Q: Show that for every positive

A:

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: Prove that 3 divides n³ +2n whenever n is a positive integer.

A: Introduction: Consider a starting point for which the assertion is correct. The proposition is true…

Q: Use induction to prove that for every integer 15"+2 + 162n+1 is divisible by 241. n > 0,

A: To use mathematical induction to prove the given statement

Q: n any positive integer n, there exist n consecutive composite integers

A: Given that given any positive integer n, show that there exist n consecutive composite integers.

Q: 2. Prove that the difference of an odd integer and an even integer is odd.

A: Prove that the difference of an odd integer and an even integer is odd

Q: For all integers n> 0, 32n – 1 is divisible by 8.

A:

Q: . Prove, using mathematical induction, that 13n – 8n is divisible by 5, for n ≥ 1.

A: Let us consider the statement is  P(n) : 13n - 8n is divisible by 5. For n = 1, 13n - 8n = 5 is…

Q: 6. Prove that 2n +5 and 3n+7 are relatively prime for every integer n.

A: What is Relatively Prime Numbers: Relatively prime or coprime numbers are two such numbers which…

Q: Prove that there is no integer n for such that 5n+3 is divisible by 5.

A:

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

A:

Q: prove that it ņ is an 3n +2 is odd then integer and is add

A: This is related to Number System

Q: Prove that 4 divides 6 7" - 2.3" for all integers n 20.

A: We need to prove that 4 divides 6.7n-2.3n for all integers n≥0.

Q: Prove that n3+(n+1)3+(n+2)3 is divisible by 9 for all non-negative integers  n.

A:

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

A:

Q: Use proof by cases to prove that for any integer n, n² + 3n + 7 is even.

A:

Q: - Find every positive integer n with the property that 2" - 1 is divisible by n.

A: