Paper 4, Section II, 6E6 \mathrm{E}

Numbers and Sets | Part IA, 2016

Suppose that a,bZa, b \in \mathbb{Z} and that b=b1b2b=b_{1} b_{2}, where b1b_{1} and b2b_{2} are relatively prime and greater than 1. Show that there exist unique integers a1,a2,nZa_{1}, a_{2}, n \in \mathbb{Z} such that 0ai<bi0 \leqslant a_{i}<b_{i} and


Now let b=p1n1pknkb=p_{1}^{n_{1}} \cdots p_{k}^{n_{k}} be the prime factorization of bb. Deduce that ab\frac{a}{b} can be written uniquely in the form


where 0qi<pini0 \leqslant q_{i}<p_{i}^{n_{i}} and nZn \in \mathbb{Z}. Express ab=1315\frac{a}{b}=\frac{1}{315} in this form.

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