It is difficult to believe that here comes our last class. Today we had our last final exam (first online test). Please find the sample paper with answer key for your reference:)

Please read it through and feel free to reach out if you have any questions. The course ends, but our friendship lasts forever:)

(Why everyone seemed to be so “serious” after our exam ：P)

Enjoy a great summer!

Lesson review
Problem Solving Strategies:

• change the forms of numbers
• understand different solutions/methods
• parity considerations (modulo 2)
• narrow your search to a small list
• use clues to gather enough information to solve the problem

Homework:

Page 297-299

Ex.15.22, 15.24, 15.28, 15.31, 15.34, 15.35, 15.40, 15.44,  15.48, 15.49, 15.56, 15.61

Prepare for the final exam next week!

• (Login to Google Classroom at 12:20 pm–the test paper will be released then).
• Print them out, and do it in our Zoom class — keep your camera on all the time (test will end at 1:50 pm. Make sure you turn in it by 2:00 pm)

Get well prepared and Good luck!

Lesson review:

• a linear congruence equation
• an inverse: b*b^(-1) is congruent to 1 (mod m)
• only if gcd (m,r)=1, otherwise r has no reverse modulo m.
• if ak is congruent to bk (mod mk), then a is congruent to b (mod m)
• if ac is congruent to bc (mod m), then a is congruent to b (mod m/gcd(m,c))
• many remainder problems are just linear congruences.

Extra Resource

• https://mathworld.wolfram.com/LinearCongruenceEquation.html
• https://www.youtube.com/watch?v=4-HSjLXrfPs
• https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Book%3A_Elementary_Number_Theory_(Raji)/03%3A_Congruences/3.03%3A_Linear_Congruences

Homework

• Pg. 270 Ex.14.3.1, 14.3.3, 
• Pg. 275 Ex.14.4.1, 14.4.4, 14.4.5
• Pg. 278 Ex. 14.20, 14.27, 14.29  to 14.31

Feel free to reach out if you have any questions in this Chapter. We will learn Ch.15 next Sunday, and have our final exam on 6/21.

Make sure you study hard and start your review as early as possible:)

Sorry about the technical issue that I could not connect my iPad to Zoom in class today. Later on, I downloaded the newest version of Zoom and was able to do it. Here are the screenshots of the problems on my “TinyBlackboard”. I hope everyone got a clear understanding of the first part of Ch.14.

Lesson Review:

• an inverse: b*b^(-1) is congruent to 1 (mod m)
• only if gcd (m,r)=1, otherwise r has no reverse modulo m.

Extra Resource:

• https://mathworld.wolfram.com/LinearCongruenceEquation.html
• https://www.youtube.com/watch?v=4-HSjLXrfPs
• https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Book%3A_Elementary_Number_Theory_(Raji)/03%3A_Congruences/3.03%3A_Linear_Congruences

Homework:

• P260 Problem 14.1-14.5
• P264, Ex 14.2.1-14.2.3

Plan for the following weeks:

• 6/7: Ch.14 (II)
• 6/14: Ch.15
• 6/21: Final Exam

Congratulations to Jason, Julia, Alex, Susan, and Gioia on the Kahoot! Game today:)

Lesson review:

• Let m be a nonnegative integer. Every nonnegative integer is congruent modulo 2^m, modulo 5^m, and modulo 10^m to its last m digits.
• Every nonnegative integer is congruent modulo 3, modulo 9 to the sum of its digits.
• Every nonnegative integer is congruent modulo 11 to the alternative sum of its digits.

• 2n⇋units digit is even
• 5n⇋units digit is 5 or 0
• 10n⇋units digit is 0

• k*2^m ⇋ its last m digits themselves form an integer that is a multiple of 2^m
• k*5^m ⇋ its last m digits themselves form an integer that is a multiple of 5^m
• k*10^m ⇋ its last m digits are all 0.

• 3n⇋ the sum of its digit is 3k
• 9n⇋ the sum of its digit is 9k
• 11n⇋ the alternating sum of its digit is 11k

Extra Resource:

• Divisibility rules for 2-12

• Divisibility rule for 11

• Divisibility rule for 9

Homework:

• Pg. 252 Ex.13.2.1 (a,c), 13.2.2(c,f), 13.2.3(a,f), 13.2.4
• Pg. 255 Ex.13.3.2, 13.3.4, 13.3.6
• Pg. 257 Ex. 13.15 to 13.19, 13.25

No School next week. Enjoy the Memorial Day Holiday, and see you all on May 31st!

Lesson review

Summary

• Modular system, modulus m, residue r
• r is the modulo-m residue of n when n\equiv r (mod m) and  0<= r < m
• congruent or equivalent modulo m
• The integers congruent to r modulo m make up a residue class (congruence classes or equivalence classes).

Extra Resource

• https://artofproblemsolving.com/wiki/index.php/Modular_arithmetic/Introduction
• https://study.com/academy/lesson/modular-arithmetic-rules-properties.html
• https://betterexplained.com/articles/fun-with-modular-arithmetic/
• https://www.khanacademy.org/math/ap-calculus-bc/bc-series-new/bc-series-optional/v/deriving-geometric-series-sum-formula

Homework

• p.221 Ex.12.2.1 (i,j,k,l), 12.2.3, 12.2.6
• p.224 Ex.12.3.1, 12.3.2 to 12.3.4(a,c,e,f)
• p.230 Ex.12.4.1 (a,c,e,f),12.4.2, 12.4.4
• p.236 Ex.12.5.1(a,c,e,f), 12.5.2, 12.5.5
• p.239 Ex.12.6.2(a), 12.6.5
• p.242 Ex.12.26

Sorry that we have so much information for this chapter 🙁 Please feel free to reach out if you have any questions. See you next Sunday!

Some students asked me about this problem. Here is the answer:

Make sure you have considered all the possibilities:)

Ch.11 Decimals and Fractions

Congratulations to Annabelle, Susan, and Jason on the Kahoot! game!

Lesson review:

• terminating decimals
• repeating decimals (recurring decimals)
• radix point: the dot to separate an integral part of a number from the fractional part
• divide by 10^n — move the decimal point n places to the left
• a/b in lowest term: if b has a prime factor other than 2 or 5, then a/b has a repeating expansion
• a/b in lowest term: # of digits past the decimal point is the larger of the power of 2 and 5 in the prime factorization of b.
• 3 types of decimals:
  • Terminating decimals: have a finite number of digits after the decimal point.
  • Recurring decimals: have one or more repeating numbers or sequences of numbers after the decimal point, which continue infinitely.
  • Irrational numbers: decimals which go on forever, never- ending and never forming a repeating pattern; cannot be expressed as a fraction. 

Extra Resource:

• https://www.futurelearn.com/courses/maths-subject-knowledge-fractions-decimals-and-percentages/0/steps/66340
• https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-numbers-operations/cc-8th-repeating-decimals/v/converting-a-fraction-to-a-repeating-decimal
• https://www.youtube.com/watch?v=uSf3fl_Ho70

Homework:

• p.198 Ex.11.2.1 (a,c,f,h), 11.2.2 (a,b,c,d)
• p.203 Ex.11.3.1 (a,c,f,h), 11.3.2
• p.206 Ex.11.4.1 (a,c,e)
• p.210 Ex.11.5.1(a,c,f,h), 11.5.2 (a,b,c,d)
• p.211 Ex.11.19, 10.20
• p.212 Ex. 11.26, 11.29(b), 11.30

See you next Sunday!!!

Welcome new friend Gioia! ^_^

Congratulations to the Top 3 in Kahoot! game today: Max, Susan, Annabelle ^_^

Today we learned Ch.10 Units Digits. Here are the blackboard notes in class for your review.

Homework:

• p.182 Ex.10.2.1 (h,m,q,t), 10.2.4 to10.2.6
• p.185 Ex.10.3.1 (c,g,h,i), 10.3.3
• p.188 Ex.10.4.1, 10.4.3, 10.4.4 (a)
• p.189 Ex.10.20, 10.23, 10.27
• p.190 Ex.10.37, 10.41

Extra Resource:

• https://www.justquant.com/numbertheory/units-digit-of-a-number-raised-to-power/
• https://www.youtube.com/watch?v=q2P8KxG3FHk
• https://encrypted-vtbn0.gstatic.com/video?q=tbn:ANd9GcTmIbL2kv7wDSrKeD718sxOXcbaLFUR-0NWEUG2gNqhL_xEqprJ
• https://artofproblemsolving.com/wiki/index.php/1997_AJHSME_Problems/Problem_25

Summary：

• For +,-,X of positive integers, the units digit of the result is determined solely by the units digits of all the integers involved. 
• However, the division can not apply this rule.
• Problem-solving: hunt for patterns 🙂

Take care and see you next week!

Ch. 9 Base Number Arithmetic

We learned +,-,*,/ of base numbers today, and played another Kahoot! game.

Congratulations to the Top 3 students: Susan, Max, and Jason!

Extra Resource:

• http://core.ecu.edu/csci/wirthj/Basen/bnArith.html
• https://www.youtube.com/watch?v=U-19aiF2pbg
• https://www.youtube.com/watch?v=vIAOtiRkxDc
• https://demonstrations.wolfram.com/BasicArithmeticInOtherBases

Homework:

• p.165 Ex.9.2.1 (b,d,f,j)
• p.168 Ex.9.3.1 (a,d,e,i)
• p.170 Ex.9.4.1 (a,d,f）
• p.172 Ex.9.5.1(a,d,b,e), 9.5.2, 9.5.4
• p.173 Ex.9.15 (d), 9.16 (e)
• p.174 Ex.9.17

I have sent the feedback to the Bonus Test via email. Please check it (no worry about the score, as we have only 6 questions, and it only counted the final answer, without giving credit to your process), and feel free to reach out if you have any questions. (We may take the final exam online this semester. Most likely I will design another Google Form:P )

Take care and see you next week. We will learn Ch. 10 Units Digits then!