Let n=pp Z+ where p₁ and p2 are distinct prime numbers and 7₁,72 ≥ 1. If o(n) = 80, then determine all such positive integers n.

Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: З and an - - 3ап-1 + an-2 for integers n>2. Evaluate and d4: Let a, = 2, a2

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

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Q: Find all integers 100 ≤ n ≤ 999 such that the last three digits of n² is same as n.

A: To find: All integers  such that the last three digits of is same as n.

Q: determine the value of c(n,5) /p(n,5) where n can be any integer thats 5 or greater

A: Recall the following. Permutations Number of permutations of n things taken r at a time: pn,r=n!n-r!…

Q: For each nonnegative integer i, what is the least residue modulo 9 or 10i?

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Q: 5. Show that 2 32 3" 31 3" for all positive integers n.

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Q: nd the smallest positive integer N such that n+2 3n-4 -}| < < 0.01 for all n ≥ N.

A: We want to find N such that n+23n-1 - 13 &lt; 0.01    for all   n ≥ N

Q: Prove that p=5 and q=2 are the only pair of prime numbers such that p−q=3.

A: you can solve it by using the solution of Diophantine equation. So , Here we have to prove  p = 5…

Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

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: 3. Let a₁ = 2, a2 = 1, and for each natural number n > 2, let an 2an-1-an-2. Prove that for all…

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Q: Show that any positive integer n with n ≥ 4 can be expressed as n = 2x + 5y where x ≥ 0, y ≥ 0.

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Q: Let n = pp2 Z+ where p₁ and p2 are distinct prime numbers and T1, T21. If o(n) = 60, then determine…

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Q: Show that yn for all n=1, 2₁- ✓

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Q: p, q, pq | (apq − ap − aq  + a) for all a.

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Q: Let ß = (1,3,5,7,9,8,6)(2,4,10). What is the smallest positive integern for which ßn = ß-5?

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Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: For each positive integer n, show that и(п)и(n + 1)u(п + 2)u(n + 3) — 0

A: To prove : μ(n) μ(n+1) μ(n+2) μ(n+3) = 0  for each positive integer n.

Q: Let N=20. Show using the Sieve of Eratosthenes how to find all prime numbers between 10N and 10N+100…

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Q: Show that if 2n−1 is prime then n is prime.

A: We have to show that if 2n-1 is prime then n is prime.

Q: Use proof by contradiction to show that the sum of two odd integers(k and l) is eve

A: An even number is a number that is divisible by two. All even numbers finish with one of the digits…

Q: Let x and y be positive integers. When is x + y3 a prime number? When is x+y a composite number?…

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