OPC 2 Problem 2

$2. \text{Prove that no number in this sequence } 11,111,1111,11111,........... \\ \text{ is a perfect square}$

Give different methods of solving this. This was a problem to the OPC 2. The OPC 2 has already ended.

#Proofs #Contests #Problems

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Don't get me wrong, but I feel like most of the problems are from "an excursion in mathematics"

Vaibhav Prasad - 6 years, 2 months ago

Well I have heard a lot about that book. But these problems I have thought up and one is a famous one from IMO.

Rajdeep Dhingra - 6 years, 2 months ago

OK Allright :)

Vaibhav Prasad - 6 years, 2 months ago

write

$\underbrace {1 \cdots 1} _ {n \textrm{ times}}= a^2$

Of course $a$ must be odd (and greater than $1$ since our sequence begins with $11$ , in other words $n \geq 2$ ). Consider the even $b = a -1$ . We have

$\underbrace {1 \cdots 1} _ {n \textrm{ times}}= b^2 -2b +1$

$\underbrace {1 \cdots 1} _ {n-1 \textrm{ times}}\times 10= b^2 - 2b$

$b$ is even so $4$ divides the LHS, but clearly not th RHS. Contradiction.

Andrea Palma - 6 years, 2 months ago

The numbers in the sequence can be written as $111....108+3$ now since the last two digits, i.e. $08$ , is divisible by $4$ , the numbers are of the form of $4k+3$ . Now note that a perfect square, say $a^2$ when written in modulo $4$ $\implies a^2=0,1,2\pmod4$ but not $=3\pmod4$ which contradicts to the form of the numbers in the sequence.

Marc Vince Casimiro - 6 years, 2 months ago

any no will have last two digits 11 and therefore cannot be a perfect square .i hope u get it .if u dont just ask

Kaustubh Miglani - 5 years, 5 months ago

You mean any number with last 2 digits 11, 31, 51, 71, 91 cannot be a perfect square?

Andrea Palma - 5 years, 5 months ago

Yup, It can't be.

Rajdeep Dhingra - 5 years, 5 months ago

absolutely.

Kaustubh Miglani - 5 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$