Is the product of two odd integers odd or even Give a comple

Is the product of two odd integers odd or even? Give a complete justification for your answer Question 10. Prove the following statement:

I am solving Q9 and Q10, please post one more question to get the remaining two answers. Thanks

Q9)

Let the two odd integers be a and b, then a and b can be written as

a = 2p + 1

b = 2q + 1

where p and q are integers

a * b = (2p+1) * (2q+1) = 4pq + 2p + 2q + 1 = 2(2pq+p+q) + 1

2pq + p + q, will be a new integer P, then ab = 2P + 1, hence the product of two odd numbers will always be an odd number

Hence proved

Q10)

Proof by using mathematical induction

Base Case: (n=1)

LHS = 1

RHS = 1^2 = 1

Hence the base case holds true

Let us assume that the given thing holds true for n=k

1 + 3 + 5 + ... + (2k-1) = k^2

Now we need to prove that given thing holds true for (n=k+1)

LHS = 1 + 3 + 5 + .... + (2k-1) + (2(k+1) - 1)

using the above factor that (1 + 3 + 5 + .... + (2k-1)) = k^2, we get

LHS = k^2 + 2k + 2 - 1 = k^2 + 2k + 1 = (k+1)^2

RHS = (k+1)^2

Hence by the principle of mathematical induction we get

1 + 3 + 5 + .... + (2n-1) = n^2