7. Suppose that m, n E N are two consecutive numbers such that both m > n > 2. Prove that if n is a prime number, then m must be composite.

Q: Calculate C(12,5) by hand. As C(n,r) = n!/(n-r)!r! identify n and r, and write out every number in…

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Q: Suppose that A, B, C, and D are digits from 1 to 9 and that A + B = CD (a) What digit must C be?…

A: Given that  A, B, C and D are digits from 1 to 9 such that  A + B = CD   (a)  If 1 ≤ A, B ≤ 4 then A…

Q: 3. Let, a, = 3, a2 = 4 and for n2 3, a, = 2an-1+ an-2 + nº+ 1, express a, in terms of n.

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Q: 5. Prove that, for n> 1, n

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Q: 17. Prove that n is square - free every prime number to sat:sfies does not divide ).

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

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Q: 6. Prove that n, n+2, n +4 are all primes if and only if n = 3.

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Q: 4. Prove that if n is an integer and n is a multiple of three (that is, n = 3k for some integer k),…

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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: The sum of the first n natural numbers, S = 1 + 2 + 3 + first n natural numbers is 171, determine…

A: Consider the given information:

Q: Let h be a prime number and let k be some integer. What are the possible values of ged(h+k, h-k)?

A: We have given that ,  h be any prime number and k be some integer. We need to find possible values…

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: 5. Prove that (n, n + 2) = 1 is n is odd and (n, n + 2) = 2 is n is even.

A: Suppose (n, n+2) = d => d | n and d | n+2 => d | (n+2) - n => d | 2 => d = 1 or 2.

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: 1. Prove that f1 + f²+...+ ƒ/²/2 = ƒnƒn+1, where n is a positive integer and fn is the nth Fibonacci…

A: The Fibonacci numbers are the numbers in the following integer sequence.…

Q: 2. Find n and a if: Inx² + In2= In ax"+1

A: Given equation is lnx2+ln2=lnaxn+1. We have to find the value of n and a. Used formula: lnmn=nlnm…

Q: 10. Let a and b be natural numbers that are co-prime. Prove that (b-a) and b must also be co-prime.

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Q: Show that yn for all n=1, 2₁- ✓

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Q: Let p and q be primes, Prove that plq if and only if p = = 9.

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Q: 3. Prove that when you select n+1 numbers from {1,2,...2n} two of them Must be consecutive (that is,…

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Q: 2. Prove that if n is even, then 3n + 1 is odd

A: The given expression is 3n + 1. Prove that 3n + 1 is odd, if n is even.

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: Exercise 1. A student is asked to prove If m is even and n is odd, then m² + n² is odd. 2 The…

A: Let us answer above questions

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: 24. Suppose that I choose six of the first ten positive integers. Prove that I must have chosen two…

A: Rewrite the first ten numbers as follows.

Q: Exercise 1.8. Prove that if one chooses n+1 numbers from {1, 2, 3,...,2n}, it is guaranteed that two…

A: Prove that if one chooses n+1 numbers from the set {1,2,3,…,2n}, then two numbers will be…

Q: Show that if a and b are relatively prime then (a+2b,2a+b)=1 or 3.

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