Question
Sometimes, division by 648 is actually multiplication by 12. Answer the following about modular arithmetic, for 10 points each.
[10m] For a system of integer congruences each of the form “x is congruent to a-sub-i mod n-sub-i” with coprime moduli, this theorem ensures an integer solution.
ANSWER: Chinese remainder theorem [accept CRT]
[10h] The Chinese remainder theorem generalizes to rings by replacing the moduli with these sets and coprimality with comaximality. Two of these things are comaximal if 1 can be written as a sum of elements chosen from them.
ANSWER: ideals [accept pairwise comaximal ideals]
[10e] To actually solve a system of integer congruences, one uses repeated division in Euclid’s algorithm for computing this function. Two integers are coprime if this function returns 1 when applied to them.
ANSWER: greatest common divisor [accept greatest common factor or gcd or gcf]
<RA, Other Science: Math>
Summary
| 2023 ARCADIA at Duke | Emory, Duke, Yale | Y | 4 | 5.00 | 50% | 0% | 0% |
| 2023 ARCADIA at Emory | Emory, Duke, Yale | Y | 4 | 12.50 | 50% | 25% | 50% |
| 2023 ARCADIA at Imperial | Imperial | Y | 5 | 18.00 | 80% | 60% | 40% |
| 2023 ARCADIA at Maryland | Maryland, Online | Y | 3 | 13.33 | 67% | 33% | 33% |
| 2023 ARCADIA at Ohio State | Ohio State, Texas | Y | 3 | 6.67 | 33% | 33% | 0% |
| 2023 ARCADIA Online | Maryland, Online | Y | 3 | 16.67 | 67% | 67% | 33% |
| 2023 ARCADIA at Texas | Ohio State, Texas | Y | 3 | 13.33 | 67% | 33% | 33% |
Data
| Edinburgh | Birmingham | 0 | 0 | 10 | 10 |
| Warwick | Cambridge A | 10 | 10 | 10 | 30 |
| Imperial B | Durham | 10 | 10 | 0 | 20 |
| Imperial A | Cambridge B | 10 | 0 | 10 | 20 |
| Oxford | Bristol | 0 | 0 | 10 | 10 |