Too many ones

Number Theory Level 1

Find the square of the number $111,111,111$ .

Just to clarify: the number is one hundred eleven million, one hundred eleven thousand, one hundred eleven.

The answer is 12345678987654321.

2 solutions

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Anna Anant
Jan 6, 2015

Repeated 1 is a special number that if multiply by itself will result in a number sequence from 1 to n (where n is the number of 1) and count down to 1.

E.g. 1111*1111 = 1234321 because there are four 1. But this rule only applies if the number of 1 is less than 10. So 111,111,111 squared is 123456789 and then count down to 1 12345678987654321

Chew-Seong Cheong
Dec 29, 2014

This can be solved by induction as follows.

$\begin {matrix} 11^2 & = & 121 \\ 111^2 & = & 12321 \\ 1,111^2 & = & 1234321 \\ 11,111^2 & = & 123454321 \\ 111,111^2 & = & 12345654321 \\ 1,111,111^2 & = & 1234567654321 \\ 11,111,111^2 & = & 123456787654321 \\ 111,111,111^2 & = & \boxed{12345678987654321} \end {matrix}$

Moderator note:

Poor choice of phrasing. Your implication of "induction as follows" suggests that $1,111,111,111^2 = 12345678910987654321$ , $11,111,111,111^2 = 123456789101110987654321$ , and so on. Which is clearly wrong. You should mention this pattern only works when the number is in the form $\underbrace{111 \ldots 1}_{n \ 1\text{'s}}$ for $n=1,2,3, \ldots , 9$ only.

I was actually more concerned about typing the answer correctly than typing the correct answer lol :)

Wee Xian Bin - 6 years, 5 months ago

Too many ones

The answer is 12345678987654321.

2 solutions

Discussions for this problem are now closed

Moderator note:

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