# 雅楽多文書館

2013年07月の記事一覧

## Congruences

Exercise
Find all incongruent solutions to each of the following congruences.
(a) $7x \equiv 3 \, ( \!\!\!\mod 15)$
(b) $6x \equiv 5 \, ( \!\!\! \mod 15)$
(c) $x^2 \equiv 1 \, (\!\!\!\mod 8)$
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## Infinite Product

Exercise
Calculate
\begin{equation*}
\prod_{k=1}^{\infty}\frac{2k-1}{2k}
\end{equation*}

Solution
http://kiyosihp.web.fc2.com/infproduct.pdf

## M-World

Exercise
Let M be a set of positive integers that are congruence to 1 modulo 4. And we say that $n$ is an M-prime if its only M-divisors are 1 and itself.
(a) Find the first six M-primes.
(b) Find an integer $n \equiv 1 (\mod 4)$ that has two different factorizations as a product of M-primes.

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