Q: Is 211,101,111 a Perfect Square?

Why is 211,101,111 not a perfect square?

A perfect square is a number that can be expressed as the product of two equal integers.

The only way to accurately calculate if a number is a perfect square is to find the factors. Before we go through the trouble of finding the factors, there is a quick trick you can use to help determine if you need even need to do the extra work.

Try these steps first:

1. A number that is a perfect square never ends in 2, 3, 7 or 8. If your number ends in any of those numbers, you can stop here because your number is not a perfect square.
2. Obtain the digital root of the number. The digital root essentially is the sum of all of the digits. If you're lost, don't worry, we'll go over each step in more detail below.
3. All possible numbers that are a perfect square have a digital root of 1, 4, 7, 9.

Let's try it...

Step 1:

What is the last number of 211,101,111? It is this number: 211101111. The answer is 1. Is 1 in the list of numbers that are never perfect squares (2, 3, 7 or 8)?

Answer: NO, 1 is not in the list of numbers that are never perfect squares. Let's continue to the next step.

Step 2:

We now need to obtain the digital root of the number. Here's how you do it:

1. Split the number up and add each digit together:

What is the digital root of number 211,101,111?

Answer: 9

Step 3:

So now we know the digital root of 211,101,111 is 9. Is 9 in the list of digital roots that are always a square root (1, 4, 7 or 9)?

Answer: YES, 9 is in the list of digital roots that are always perfect squares. We can conclude that 211,101,111 could be a perfect square!

Factoring

OK, so now we know that 211,101,111 could be a perfect square. We have to find the factors of the number to be sure.

Here are all of the factors of 211,101,111:

1 x 211,101,1113 x 70,367,0379 x 23,455,67913 x 16,238,54739 x 5,412,84947 x 4,491,513117 x 1,804,283141 x 1,497,171169 x 1,249,119423 x 499,057507 x 416,373611 x 345,5011,521 x 138,7911,833 x 115,1672,953 x 71,4875,499 x 38,3897,943 x 26,5778,859 x 23,829

Whoa! That is a lot of factors!

We're looking for a factor combination with equal numbers for X and Y (like 3x3) above. Notice there isn't a equal factor combination, that when multiplied together, produce the number 211,101,111. That means 211,101,111 is NOT a perfect square.

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