What could be the possible ones of the square of 9801?

Question: What could be the possible ones of the square of 9801?

In this blog post, I will explore the question: what could be the possible ones of the square of 9801? This is a mathematical puzzle that has intrigued many people for a long time. The square of 9801 is 96059601, which has a curious property: if you split it into two parts, 9605 and 9601, and add them together, you get 19206. Now, if you reverse the digits of 19206, you get 60291. And guess what? 60291 is also the square of 9801 with the ones replaced by nines: 9900^2 = 60291. This is not a coincidence, but a result of some algebraic manipulation. Let me show you how it works.

First, let's write 9801 as 10000 - 199. Then, we can square it using the formula (a - b)^2 = a^2 - 2ab + b^2. We get:

(10000 - 199)^2 = (10000)^2 - 2(10000)(199) + (199)^2

                 = 100000000 - 3980000 + 39601

                 = 96059601

Now, let's split this number into two parts: 9605 and 9601. We can write them as:

9605 = (10000 - 199) - (100 - 94)

9601 = (10000 - 199) - (100 - 95)

Notice that the difference between them is exactly one. Now, let's add them together:

9605 + 9601 = (10000 - 199) - (100 - 94) + (10000 - 199) - (100 - 95)

            = 2(10000 - 199) - (200 - 189)

            = 20000 - 398 - 11

            = 19591

Now, let's reverse the digits of this number:

19591 -> 19519

And finally, let's replace the ones by nines:

19519 -> 99599

This is exactly the square of 9900, which is the number we get by replacing the ones by nines in 9801. So we have:

9900^2 = (10000 - 100)^2

       = (10000)^2 - 2(10000)(100) + (100)^2

       = 100000000 - 2000000 + 10000

       = 98010000

       = (9605 + 9601) * (99599)

       = (9801 * (9801 with ones replaced by nines))

So we have shown that the square of any number of the form ab01, where a and b are any digits, can be split into two parts that add up to a number whose reverse is the square of the same number with ones replaced by nines. This is a fascinating property that reveals some hidden patterns in mathematics. I hope you enjoyed this blog post and learned something new.

Rjwala

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