## Exercises 6.7 Review Exercises

###### Writing Based Exercises

###### 1.

Prove that there are no integer solutions to \(14x \equiv 1 \pmod{21}\) using a definition of modular arithmetic.

###### 2.

There is a bucket of 15 letters. These letters include A, B and are either red or blue. There are 3 red As, 6 blue As, 4 red Bs, 2 blue Bs. Suppose we draw two letters randomly from the bucket, both at the same time. We want to calculate the probability that one letter is an A and the other letter is red.

- Explain why there is overlap in this case and why that means \(\frac{C(9,1)*C(7,1)}{C(15,2)}\) is NOT the correct probability.
- Give the correct probability.

###### 3.

Use induction to prove the following statement for all \(n\text{.}\)

###### 4.

Prove that if \(e,d\) satisfy the number trick mod \(p\) for any odd prime \(p\) then neither \(e\) nor \(d\) can be even. Remember that the number trick means \((x^e)^d \equiv x \pmod{p}\text{.}\) Does the same result hold \(\pmod{pq}\) for two distinct primes instead of \(\pmod{p}\text{?}\)

###### 5.

Consider the statement: if *@7*6@7 is prime, then dragons are orange.

- Explain in what situation the above statement would be false.
- Give the converse of the statement. What situation would make the converse false?
- Give the contrapositive of the statement. What situation would make the contrapositive false?

###### 6.

For \(n=8,16,32,64\ldots\) find the smallest \(m\) such that \(x^{1+km} \equiv x \bmod{n}\) for all of the odd \(x\) mod \(n\text{.}\) Remember this will not work for all of the \(x\text{,}\) just the odd ones.

- Previously we had some Sage code to find this for all \(x\text{.}\) Explain what you need to modify in the code below to just check this for odd \(x\text{.}\)
- Calculate the \(m\) for a number of enough different values of \(n\) until you can make a conjecture.
- Bonus: Use induction to prove the above conjecture.

###### Reflective questions

###### 7.

What did you learn (or re-learn) about making mathematical conjectures in this class? What did you learn (or re-learn) about making mathematical conjectures in this class?

###### 8.

Describe a problem (homework or in-class activity) in this course that you struggled to understand and solve, and explain how the struggle itself was valuable. In the context of this question, describe the struggle and how you overcame the struggle. What is the importance of struggle in this case? How might it benefit you in the future?

###### 9.

Many of the exercises in the homework and in-class investigations are designed to allow you to discover mathematical ideas and result for yourself. Describe a time when you felt successful at discovering a result for yourself in this class. How is it different to learn a topic by discovering it yourself versus learning about it in a lecture? Why is it important to discover it yourself?

###### 10.

Describe you favorite topic in the class and explain why it was interesting to you. What mathematical ideas are you curious to know more about as a result of taking this class? Why?

###### Cipher Challenge Activity

###### 11.

This is the first cipher for your review Cipher Challenge. It is an easy cipher.

HVS BSLH HKC QWDVSFG OFS PSZCK HVSM OFS OTTWBS CF JWUSBSFS (CBS CT SOQV).

###### 12.

Cipher 1: TDCHJ NDMJM KDHVU KDQHK LCCDY ZDMKV BHKLC CDYZD HJYML VYBLY XNWDQ JOHVU KDQBD LHKHV UKDQK LBVYO JQNLM VJYMK LMTVC CMDCC PJXKJ TMJSD HQPUM MKDYD EMHVU KDQLY SZVID PJXMK DCJHL MVJYJ OMKDY DEMHV UKDQP JXHLY LHHDB BLCCJ OMKDH VUKDQ BVYMK DMDEM WJJRM KDYDE MHVUK DQVBC JHLMD SVYBL ZDHJN UXMLM VJYMT JUJVY MYVYD UJVYM BVEVM VBLRD PTJQS HJCXN YLQHV UKDQP JXTVC CYDDS MJSDH QPUMH VUKDQ MTJMJ ZDMVY OJQNL MVJYL WJXMM KDRDP TJQSO JQMKD RDPTJ QSHJC XNYLQ HVUKD Q

Cipher 2: PVVUA MNYNR WYNHY OJSOD YWGRA FZCPV VPQZC XWAVP EFJDW BUVWG KPKFK RADVB OCEGD CJKER KREGH EKHVK XCLDY FPZPC XBWDY IYWGR AFNYN YNSHZ YKRSC SPHCO LSIWW BVXPO KDNOT DECEP KFFXA KYYAL GOYHJ BAGLV PGRDK BTOSW KRKIK VWPFB BCIDD SIOEG RFWGK KICLX PCWZQ FVVUF FEPWE OHOSY NWEDD SGBAD RBWHZ YJCWP NSHEA BTIPO SVAGK RAFVK NFRXC SDOJH FPYWG RAFWY NSOKI WEKPW FXWBU DDSKB EOCKJ RWSPH ZXCCW VAHKO NHFVA HKONP VPKFV DDSDO OGRQA PVQEB JDKOG ZAOID DWJAQ CKOWD GOWFJ SJMFE NHVHP PFYGW EDDSS KOWTS ZSRCO STDEC EYBQY KLHVB KBVIK INSHZ RVOCE OARKY ZSTBU DKMED YONCE OPCXO PHYON SJDKT KRAWE PKFDK PWFXW PFEPH YOJSO DYWGR AF