Question

Существует ли такая позиционная система счисления, в которой число 71 (записанное в данной системе счисления) является квадратом целого числа, записанного в этой же системе счисления?

Answer 1

Используем запись числа 71 по основанию n а развернутом виде.
$71_n=7\times n+1; \ a^2=7n+1 \to a= \sqrt{7n+1}$
Здесь a - некое число, квадрат которого равен 71.
Поскольку в записи числа 71 присутствует цифра семь, то система счисления в качестве основания может использовать число, не меньшее восьми.
Для n=8 под квадратным корнем получаем 7х8+1=57, корень из 57 не целый.
Для n=9 получаем 7х9+1=64, а корень из 64 целый и равен восьми.
Следовательно, система счисления девятиричная.
$(8_9)^2=71_9$

Answer 2

Positional Number System and Perfect Squares

In a positional number system, a number is represented based on the position of its digits. The most common positional number system is the decimal system, where each digit's position represents a power of 10. The question asks whether there exists a positional number system in which the number 71, when represented in that system, is a perfect square of an integer also represented in the same system.

Analysis of the Given Number System

The search results provide information about various positional number systems, but they do not explicitly mention a system in which the number 71 is a perfect square of another integer represented in the same system. However, it's important to note that the search results do not cover all possible positional number systems, so it's still possible that such a system exists.

Conclusion

Based on the available information, it is not explicitly stated in the search results whether there exists a positional number system in which the number 71, when represented in that system, is a perfect square of an integer also represented in the same system. Further exploration or mathematical analysis may be required to determine the existence of such a system.