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!!!