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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.

Note by Vishnu C
2 years, 6 months ago

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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 - 2 years, 6 months ago

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

Pi Han Goh - 2 years, 6 months ago

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