Square of 9's

Number Theory Level 3

How many zeros are there in the decimal representation of $999990^2$ ?

The answer is 6.

2 solutions

Jesse Nieminen
Mar 9, 2017

$999990^2 = \left(1000000-10\right)^2 = 1000000000000 - 20000000 + 100 = 999980000100$ .

Hence, the answer is $\boxed{6}$ .

Toshit Jain
Mar 5, 2017

Squares of 9's:| Let's have a look at some - 99^2= 9801 | 999^2 = 998001 | If we watch closely , in these cases, the 9 on the extreme right is decreased by 1 and is followed by as many zeroes as many 9 preceding it and these zeros followed by 1 | So, 99999^2 = 9999800001 | So , 4+2(obtained after squaring 10)=6 zeros !!

How do you know it works for ALL such numbers?

Pi Han Goh - 4 years, 3 months ago

It works well for squares of all numbers having only 9s. You may use a calculator and you will notice a similar pattern in all! Correct me if I was mistaken !

Toshit Jain - 4 years, 3 months ago

You have only shown that it's true for 99^2 and 999^2, how do you know that it works for 9999^2, 99999^2, .... and the rest?

Pi Han Goh - 4 years, 3 months ago

Because the same pattern is followed all through...You may use a calculator! For instance , 9999^2 = 9998001 , 99999^5=9999800001 , 999999^6=999998000001 ! and it goes on... I only observed the pattern...

Toshit Jain - 4 years, 3 months ago

Finding a pattern isn't proof.

Here you'll find an excellent example why you shouldn't trust your patterns without proving them:

https://www.youtube.com/watch?v=84hEmGHw3J8

Jesse Nieminen - 4 years, 3 months ago

Like I said earlier, you have only shown that it's true for 99^2, 999^2, 9999^2, 99999^2, 999999^2

But does this work for ALL the infinitely many numbers? If yes, PROVE IT!

Pi Han Goh - 4 years, 3 months ago

Square of 9's

The answer is 6.

2 solutions

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