Question

Prove that product of two odd integers is an odd integer ? In hindi

Answer 1

If you are really 1414 I am guessing you haven't had much experience with formal proofs. I'll attempt to make this as simple as possible. Let nn and kk be a positive integer. That is, we are choosing any two number from the list 1,2,3,…1,2,3,…. We know an integer is even if it is divisible by two. If a number is not divisible by two, it is odd. So now we will use a "trick" to give ourselves some odd numbers. No matter what n,kn,k we choose, it should be obvious that 2n,2k2n,2k are integers that are divisible by two. If we add one to 2n2n and 2k2k then it should also be clear that 2n+12n+1 and 2k+12k+1 are NOT divisible by two, due to the remainder of one. In this way, we have chosen two arbitrary odd integers. The arbitrariness is very important, because it means that the final result will be true for any two odd integers we start with. Now let's consider the product of our two odd integers,

(2n+1)(2k+1)=2n⋅2k+2k+2n+1=4nk+2k+2n+1=2(2nk+k+n)+1(2n+1)(2k+1)=2n⋅2k+2k+2n+1=4nk+2k+2n+1=2(2nk+k+n)+1

Next you can observe that (2nk+k+n)(2nk+k+n) is some integer, let's say m=(2nk+k+n)m=(2nk+k+n). Then we can write

2(2nk+k+n)+1=2(m)+12(2nk+k+n)+1=2(m)+1

and finally we can conclude that for any two odd integers 2k+1,2n+12k+1,2n+1 that

(2n+1)(2k+1)=2m+1(2n+1)(2k+1)=2m+1

for some integer mm. Hopefully this makes sense! Formal proofs can be challenging, especially when you are used to working with explicit numbers.