I hope to give you a flavour of some non-trivial factorization examples using the following identity:

**Example 1. Let n be a positive integer. Factorize **

**Solution. **Observe that

where , , and .

Thus, using the above factorization identity, we get the following factorization:

**Example 2. **Let a, b, c be distinct positive integers and let k be a positive integer such that .

Prove that .

**Solution. **The desired inequality is equivalent to

.

Suppose without loss of generality, that .

Then, since a, b, and c are distinct positive integers, we , and

and .

It follows that

We obtain

so it suffices to prove that or

But, ,

and the conclusion follows.

More later… Nalin