An interesting problem from CMI entrance exam.

A positive integer N has its first, third and fifth digits equal and its second, fourth and sixth digits equal. In other words, when written in the usual decimal system it has the form xyxyxy, where x and y are the digits. Show that N cannot be a perfect power, i.e., N cannot equal $a^ b$ , where a and b are positive integers with b > 1.

#NumberTheory

Easy Math Editor

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`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`	$\sum_{i=1}^3$
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Comments

Hint: factor $\overline{xyxyxy}$ , think about the factors of $10101$ .

Pi Han Goh - 6 years, 1 month ago

10101=13x37x3x7. xyxyxy=10101(10x+y). So, in order for it to be a perfect square, 10x+y>=10101. But 10x+y<100. So, it's not possible.

vishnu c - 6 years, 1 month 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}`	$\boxed{123}$