Hi, Please solve these questions as soon as possible, In your own words, briefly

Question

Hi, Please solve these questions as soon as possible,

In your own words, briefly explain why Mathematical Induction is a valid method of proof. Prove that if m, m + 1, m + 2 are three consecutive integers, one of them is divisible 3. Prove that for n an integer, n^3 - n is divisible by 3. Let m and n be integers. Prove or disprove that if m and n are divisible by 3, then m + n is divisible by 3. Let m and n be integers. Prove or disprove that if m + n is divisible by 3, then m and n are divisible by 3. 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

Solution

Back-up Theory

If n is divisible by m, then n = km, where k is an integer and conversely, if n = km, where k is an integer, then n is divisible by m.

Q3

Given m, m + 1 and m + 2 are consecutive integers, to prove that one of them is divisible by 3.

Proof:

If m is divisible by 3, the result is trivially true.

If m is not divisible by 3, then m must be of the form: (3k + 1) or (3k + 2), where k is any integer.

If m = 3k + 1, then m + 2 = 3k + 1 + 2 = 3(k + 1) => m + 2 is a multiple of 3.   

If m = 3k + 2, then m + 1 = 3k + 2 + 1 = 3(k + 1) => m + 1 is a multiple of 3.

Thus, one of m, m + 1 and m + 2 is always divisible by 3 if m is an integer. PROVED

Q4

To prove (n3 – n) is divisible by 3, where n is an integer.

Proof:

n3 – n = n(n2 – n) = n(n - 1)(n + 1) or = (n - 1)n(n + 1).

Given n is an integer, (n - 1), n and (n + 1) are 3consecutive integers.

So, as already proved, under Q3, one of (n - 1), n and (n + 1) is divisible by 3

=> (n3 – n) is divisible by 3. PROVED

Q5

Given integers m and n are divisible by 3, to prove/disprove that (m + n) is divisible by 3.

Proof:

m and n are divisible by 3 => n = 3k1 and m = 3k2, where k1 and k2 are integers.

=> (m + n) = 3(k1 + k2) = 3k, where k = (k1 + k2).

Since k1 and k2 are integers, k = (k1 + k2) is also an integer.

=> (m + n) is divisible by 3. PROVED

Q6

Given (m + n) is divisible by 3 where m and n are integers, to prove/disprove that m and n are divisible by 3

This is not true. So, a counter example would suffice to disprove.

Let n = 2 and m = 7. Clearly, (m + n) is divisible by 3 but neither m nor n is divisible by 3.

DISPROVED

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