A Big Deal About $9$

Number Theory Level 3

What is the sum of the digits of the number

$(9999999999)^3$

The answer is 180.

3 solutions

Ameya Salankar
May 10, 2014

$9^3=729$

Sum of the digits of $9^3$ = $7+2+9=18$ .

There are ten nines in the number.

Therefore, the sum of the digits of $(9999999999)^3 = 18 \times 10 = \boxed{180}$ .

Can you substantiate your argument? For example, the claim is not true if 9 is replaced by 2, since $2^3 = 8$ and $22^3 = 10648$ which has a digit sum of 19 ) and not $8 \times 2 = 16$ .

Calvin Lin Staff - 7 years, 1 month ago

@Calvin Lin , err... I can't find the argument. Something to do with $9$ being close to $10$ ? Looks like I was lucky (this was a competition problem)!

Ameya Salankar - 7 years, 1 month ago

Right. Do you know a better solution?

Mayank Chaturvedi - 7 years ago

Wow!!!I didn't know that this was also true.I solved it by finding a pattern.BRILLIANT.

Adarsh Kumar - 7 years, 1 month ago

9999999999=10000000000-1; (10000000000-1)^3= expand with binomial theorem; then it's easy.

Mark Kong - 7 years ago

Perfect solution. This was exactly how I solved it.

Sharky Kesa - 7 years, 1 month ago

It seems too good to be true. Can you please explain how this method works?

Shabarish Ch - 7 years, 1 month ago

Similar solution... But lucky it worked here. Nice : )

Mayank Chaturvedi - 7 years ago

Abhi Shek
Aug 21, 2014

:D 18*no. of nines :D

Vederis Leunardus
Jun 3, 2014

this is my solution hm..

99^3 = 970299 = 36
999 ^ 3 = 997002999 = 54 i see the patern hahahhaha
99 ^ 3 = 18 + 18 = 36; :)
999 ^ 3 = 27 + 27 = 54; so.. 9999999999^ 3 = 90 + 90 = 180

A Big Deal About 9 9 9

The answer is 180.

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A Big Deal About $9$