Dear Dir/Madam, Prove that if n is an integer and n^3 + 5 is odd, then n is even

Question

Dear Dir/Madam,

Prove that if n is an integer and n^3 + 5 is odd, then n is even. Prove that if n is a perfect square, then n + 2 is not. Prove that if you pick five socks from a drawer which has white, gray, blue and black socks, among them will be a pair of the same colour. Prove that if a and b are real numbers, one is greater than or equal to their average. Carefully prove that if a, b, q, r are integers, with b = aq + r, the set of common divisors of b and a is identical to the set of common divisors of a and r. Prove that if n is any integer, the only common divisor of n and 2n + 1 is 1. Prove that if n is an integer, either n^2 or n^2 - 1 is divisible by 4. Prove that the sum of the squares of two odd integers cannot be the square of an integer.

Explanation / Answer

7) if n^3 +5 is odd , then n is even

n^3 + 5 is odd hence

n^3 is even

assume that n is not even , then

n = 2k+1

n^3 = (2k+1)^3 = 8k^3 + 12 k^2 +6k +1

hence n^3 is odd

hence our assumption was wrong

hence

n is even

8) if n is perferct square, n+2 is not

n = k^2

Assume that n+2 is also perfect square

n+2 = m^2

hence 2 = m^2 - k^2

= (m-k)(m+k)

since m and k are integer

m-k = 1 , m+k = 2 as m > k

solving m = 1.5 , k= 0.5

which is absurd as m and k are integer

hence our assumption was wrong

hence n+ 2 is not perfect square

9) total 5 socks and 4 different colour

according to Pigeon-hole Principal

holes - different colours , total number of holes - 4

pigeons - different socks - total number of pigeons = 5

since 5 > 4

there will be at least a pair of same colour

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