Week 3

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Day 9

Repeating Patterns

Module 5 Day 9 Challenge Part 1 Module 5 Day 9 Challenge Part 2 Module 5 Day 9 Challenge Part 3 Module 5 Day 9 Challenge Part 4 Module 5 Day 9 Challenge Part 5 Module 5 Day 9 Challenge Part 6 Module 5 Day 9 Your Turn Part 1 Module 5 Day 9 Your Turn Part 2 Module 5 Day 9 Your Turn Part 3 Module 5 Day 9 Your Turn Part 4

Jan 5, 2022, 10:58 PM

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P Jan 5, 2022, 10:58 PM

oops made the same mistake as po-shen loh
so i learned that $\phi(mn)=\phi(m)\cdot\phi(n)$ and i applied the fact on this problem and i got 4 😞
is the phi function only multiplicative (as in the above equation) when $\gcd(m,n)=1?$
Day 10

Chinese Remainder Theorem

Module 5 Day 10 Challenge Part 1 Module 5 Day 10 Challenge Part 2 Module 5 Day 10 Challenge Part 3 Module 5 Day 10 Challenge Part 4 Module 5 Day 10 Challenge Part 5 Module 5 Day 10 Challenge Part 6 Module 5 Day 10 Challenge Part 7 Module 5 Day 10 Your Turn Part 1 Module 5 Day 10 Your Turn Part 2 Module 5 Day 10 Your Turn Part 3

Jan 8, 2022, 5:02 PM

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西 Jan 8, 2022, 5:02 PM

Okay, thanks! 🙂 @professionalbronco
Day 11

Not Relatively Prime

Module 5 Day 11 Challenge Part 1 Module 5 Day 11 Challenge Part 2 Module 5 Day 11 Challenge Part 3 Module 5 Day 11 Challenge Part 4 Module 5 Day 11 Challenge Part 5 Module 5 Day 11 Your Turn Part 1 Module 5 Day 11 Your Turn Part 2

Jan 7, 2022, 11:06 PM

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P Jan 7, 2022, 11:06 PM

Why can't we just take mod 3 to solve the "Your Turn" problem? Taking mod 3 leads to contradiction, hence solved.
Day 12

More Remainders

Module 5 Day 12 Challenge Part 1 Module 5 Day 12 Challenge Part 2 Module 5 Day 12 Challenge Part 3 Module 5 Day 12 Challenge Part 4 Module 5 Day 12 Challenge Part 5 Module 5 Day 12 Your Turn Part 1 Module 5 Day 12 Your Turn Part 2 Module 5 Day 12 Your Turn Part 3 Module 5 Day 12 Your Turn Part 4 Module 5 Day 12 Your Turn Part 5

Jan 8, 2022, 6:33 PM

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P Jan 8, 2022, 6:33 PM

@potato2017 I used the exact same method here, and it was pretty quick.

L

Week 3 Challenge Problem 19 Inquiry

• Jul 22, 2022, 4:36 AM • lucky_ducky1 Jul 22, 2022, 4:36 AM

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I don't understand the difference between relatively prime and pairwise relatively prime

• Jun 18, 2020, 7:24 PM • divinedolphin Jun 18, 2020, 7:38 PM

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D Jun 18, 2020, 7:38 PM

@divinedolphin The "pairwise relatively prime" condition is stronger than "relatively prime." For example, if two numbers have $2$ as a common factor, but the third number doesn't, then the three numbers together are relatively prime.

Relatively prime means that there is no factor common to all numbers in the group .

Pairwise relatively prime means that if you look at all possible pairs of the numbers, there is not a pair that share a common factor.

$$ \textcolor{red}{4 \text{ } \text{ } 10}\text{ } \text{ } 7 $$
$$ \text{These are relatively prime (because 7 doesn't have 2 as a factor)} $$

$$ \textcolor{red}{4 \text{ } \text{ } 10}\text{ } \text{ } 7 $$
$$\begin{aligned} \text{Look at them pairwise: } &\textcolor{red}{4} \text{ and } \textcolor{red}{10} \text{ are both multiples of 2 }; \\ &\textcolor{red}{4} \text{ and } 7 \text{ are not both multiples of some number,}\\ \text{ and } &\textcolor{red}{10} \text{ and } 7 \text{ are not both multiples of some number} \end{aligned} $$
$$ \text{These three numbers are not pairwise relatively prime. } $$

$$ \textcolor{blue}{44 \text{ } \text{ } 13 \text{ }\text{ } 25 } $$
$$ \text{ These are not all multiples of the same } x, \text{ for some number } x, \text{ so they are relatively prime} $$

$$ \textcolor{blue}{44 \text{ } \text{ } 13 \text{ }\text{ } 25 } $$
$$\begin{aligned} \text{Look at them pairwise: } &\textcolor{blue}{44} \text{ and } \textcolor{blue}{13} \text{ are not both multiples of some number }; \\ &\textcolor{blue}{13} \text{ and } \textcolor{blue}{25} \text{ are not both multiples of some number,}\\ \text{ and } &\textcolor{blue}{44} \text{ and } \textcolor{blue}{25} \text{ are not both multiples of some number} \end{aligned} $$
$$ \text{These three numbers are also pairwise relatively prime.} $$

Day 9

Day 10

Day 11

Day 12

Week 3 Challenge Problem 19 Inquiry

I don't understand the difference between relatively prime and pairwise relatively prime