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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 3

Modulo $33$,
$$
10^2=100\equiv1,
\qquad
10\not\equiv1,
$$
so $\operatorname{ord}_{33}(10)=2$. This order is even and
$$
10^{2/2}=10\not\equiv-1\pmod{33}.
$$
The <factor extraction from an even modular order> therefore gives
$$
\gcd(10-1,33)=3,
\qquad
\gcd(10+1,33)=11.
$$
Thus
$$
\boxed{33=3\cdot11.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 10D / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 10D / c / Solution

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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 10D / c / Solution

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