# FMM10

### From charlesreid1

# Friday Morning Math Problem

## Square Free Sequence

Prove that no number in the sequence

11, 111, 1111, 11111, ...

is the square of an integer.

Solution |
---|

If s is a number in the sequence, s must have the form
where m is a non-negative integer. This can be rearranged into two parts: the part divisible by 4, and the part not divisible by 4:
which means that when divided by 4, all numbers in the sequence have a remainder of 3. Furthermore, we know that all squares are of the form
or
and therefore leave remainders of 0 or 1 when divided by 4. Thus, a number s in the sequence that always has a remainder of 3 cannot be a square. |

## Flags

Friday Morning Math ProblemsSpider Socks and Shoes FMM1 Sums of Powers of 2 FMM2 Fifty Coins FMM3 The Zeta Monogram FMM4 The Cthulhus Monogram FMM4B Multiplication Logic FMM5 The Termite and the Cube FMM6 Sharing Dump Trucks FMM7 The Flippant Juror FMM8 Bus Routes FMM9 A Robust Bus System FMM9B Square-Free Sequence FMM10 Inferring Rule from Sequence FMM11 Checkerboard Color Schemes FMM12 One-Handed Chords FMM13 First Ace FMM14 Which Color Cab FMM15 Petersburg Paradox Revisited FMM16 A Binomial Challenge FMM17 A Radical Sum FMM18 Memorable Phone Numbers FMM19 Arrange in Order FMM20 A Pair of Dice Games: FMM21
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