How Many 9's?

Algebra Level 2

If $N=999,999,999,999,999,999,999$ then how many 9's are there in $N^2$ ?

The answer is 20.

2 solutions

Paco Escobar
Feb 26, 2016

We can notice a pattern if we look to: $9^2$ =81, $99^2$ = 9 801, $999^2$ = 99 8001, $9999^2$ = 999 80001

The number of 9's in N is equal to the number of 9's in $N^2$ minus 1. So if we have 21 9's in N, we should expect to have 20 9's in $N^2$ .

Moderator note:

Can you explain the mathematical reasoning behind this observation?

Hint: $9 \ldots 9 = 10 ^n - 1$ .

Can you explain the mathematical reasoning behind this observation?

Hint: $9 \ldots 9 = 10 ^n - 1$ .

Calvin Lin Staff - 5 years, 3 months ago

If the number of 9's are the same. We subtract 1 then subtract the difference from the same place value of 9. For example 99x99=(99-1) =98(9-9)(9-8) =9801 999x999=(999-1) =998(9-9)(9-9)(9-8) =998001

John Kennard Soriano - 5 years, 3 months ago

Yes, that's one way of looking at it. There's (to me) a cleaner approach that could be taken, by looking at $(10^n -1 ) ^2$ .

Calvin Lin Staff - 5 years, 3 months ago

@Calvin Lin – Thank you sir.

John Kennard Soriano - 5 years, 3 months ago

Kay Xspre
Feb 29, 2016

We wrote $N = 10^{21}-1 \Rightarrow N^2 = 10^{42}-2(10^{21})+1$

This may be alternatively written as $N^2 = 10^{21}(10^{21}-2)+1 = 10^{21}(N-1)+1$

Given that $N$ contains 21 nines, $N-1$ shall contain 20 nines, or simply expand to $\underbrace{999\dots99}_{20 \:nines}8\underbrace{000\dots00}_{20 \:zeros}1$

How Many 9's?

The answer is 20.

2 solutions

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