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Quora math question

In my math homework problem, they ask me to prove that a number which is made up with 12 ones and 13 zeros cannot be a perfect square. Can anybody help me prove it?

Answered by Sanket Alekar, math & geography nerd turned stand-up comedian

A number with 12 ones and 13 zeros has a sum of 12 when all the digits are added.

The number 12 is divisible by three, meaning that the long number is divisible by three. However, 12 is not divisible by nine, meaning that the long number is not divisible by nine.

If a number is divisible by three and not by nine, that means it has only one factor of three in its prime factorization.

12/3 = 4, 12 = 3x2x2 (only one 3)

As a result, it cannot be a perfect square, because in perfect squares each prime factor must appear an even number of times. E.g.: 3x3 = 9, 2x2x2x2 = 16

[answer edited slightly]

I would add, “It is helpful to think of more numbers, such as 27. It does not have an even number of prime factors: 3x3x3. Remember, a prime is divisible only by itself and one. Now, think of 36, a perfect square, which is 2x3x2x3. It has an even number of prime factors, two two’s and two three’s.”