Lesson review:

• factor tree
• prime factorization
• rational numbers
• lowest form (reduced form) vs. unreduced form
• gcd (a,b)
• lcm [a,b]
• ab = gcd (a,b)*lcm [a,b]
• gcd(ac,bc) = cgcd (a,b)
• lcm [ac,bc] = clcm [a,b]

Homework:

• P.68 Ex.4.2.1 (b) (k) (p)
• P.71 Ex.4.3.2, 4.3.4
• P.74 Ex.4.4.1 (b) (f), 4.4.2 (b) (d)
• P.77 Ex.4.5.1 (d) (e) (g) (h)
• P.79 Ex.4.6.1, 4.6.5
• p.83 Ex.4.7.1 to 4.7.4
• * bonus: p.86 Ex. 4.36, 4.37, 4.38, 4.39

Other Resources:

• https://www.mathsisfun.com/prime-factorization.html
• https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-prime-factorization-prealg/v/prime-factorization

Lesson Review:

• Common divisor
• GCD (Greatest Common Divisor) or GCF (Greatest Common Factor)
• Relatively prime or coprime
• Common multiple
• LCM (Least Common Multiple)
• gcd(m,n)
• lcm[m,n]
• The Division Theorem: a = bq + r (0≤r<b) a:dividend, b:divisor, q: quotient, r: remainder
• Euclidean algorithm: gcd(m,n) = gcd (m-n,n)
• The extended Euclidean algorithm: gcd(m,n) = gcd (m-n,n) = gcd (m-kn,n) = gcd (r,n)

Homework:

• Page 40: Ex 3.2.5
• Page 43: Ex 3.3.2
• Page 45: Ex 3.4.1 (d) (g)
• Page 50: Ex 3.5.3
• Page 53: 3.6.1, 3.6.6
• Page 59: 3.7.1 (b)
• Page 61: 3.24 (d), 3.32, 3.33, 3.34

Other Resources:

• https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/the-euclidean-algorithm
• https://www.khanacademy.org/math/pre-algebra/pre-algebra-factors-multiples/pre-algebra-greatest-common-divisor/v/greatest-common-divisor

Take care, and wish everyone to stay safe and healthy!

Sums of Consecutive Counting Numbers

by Mr. John Bookston

Definition:  A sum of consecutive counting numbers can start with any counting number add at least the next bigger counting number and stop adding after any number of consecutive integers.

Examples:  3+4 = 7                   7 is a consecutive sum of counting numbers

   1+2+3+4+5 = 15    4+5+6 = 15 and 7+8 = 15     15 is a consecutive sum in 3 distinct ways.

Investigation: 

For numbers that are consecutive sums, investigate patterns to predict the number of distinct ways larger numbers can be written as consecutive sums.  

Starters:       1 and 2 are not consecutive sums

Every odd integer starting with 3 can be written as a consecutive sum of two counting numbers:

                                      (n-1)/2  +  (n+1)/2  =  n       for every odd number, n.

Make a conjecture about which numbers are expressible as “the sum of 3 consecutive counting numbers”. 

Extension: Do the same for numbers that can be expressed as sums of 4,5 and 6 consecutive counting numbers.

Extension:  Consider the prime factorization of the counting numbers that cannot be expressed as a consecutive sum.  Make a conjecture as to which counting numbers less than 1000 fall into that category.

Enjoy.             

Good luck. 

Dear all,

Enclosed please find the excel file with the answer to the bank question in our brain stretcher last week. Many thanks to Mr. Zhang (Jason’s dad) for sharing!

Q4: Bank of America has money in the forms of $1, $5, $10, $20 and $100. How many different combinations can Tony withdraw $100 dollars at the counter?

https://docs.google.com/spreadsheets/d/1_GSWdR27K8XLUscFJAIIvpCTKWQFYNtYVfrSxpwv00s/edit?usp=sharing

There is no school for the following two weeks, but please feel free to contact me if you have any questions regarding our previous chapters/exercises. I will be very glad to help.

Enjoy the spring break, and I look forward to meeting you all when we are back!

Best,

Lijia

AoPS_Number_Theory 2020 Spring Week 2:  Primes and Composites

Welcome new friend Quinn! It’s great fun to try our first online class today. The kids did a great job participating in the class activities!

Lesson Review:

• Prime number & composite number
• 0,1 (special numbers)
• Methods to identify prime numbers:
  • Sieve of Eratosthenes
  • Test prime factors from 2 to root square of q
• Perfect number (a positive integer that is equal to the sum of its proper divisors): 6, 28, 496, 8128…

Homework:

• Page 31: Ex 2.3.2
• Page 35: Ex 2.4.1, 2.4.2
• Page 37: Ex 2.7, 2.9 (a) (c) (e), 2.10. 2.11
• Page 38: Ex 2.14, 2.15

Other Resources:

https://www.mathsisfun.com/numbers/prime-number-activity.html

https://www.britannica.com/science/perfect-number

No school for the following 2 weeks. No school for the following 2 weeks. Hope the Coronavirus gets controlled soon. Best wishes to our motherland China!