Workshop 2015/01/24 - Instruction topics
Factors (see bottom of page):
Questions about factors:
What does it mean if the number of factors is odd? What
does it indicate about the prime factors? Define perfect
square.
What does it mean if the number of factors is prime?
What does it mean if the number of factors is a power of 3?
How many factors does 1400 have?
What number less than 50 has the maximum number of
factors?
What is the smallest number that has exactly 16 factors?
What is the sum of all factors of a certain number?
What numbers less than 100 have the largest numbers of
factors?
Answer (12 factors each): 60,72,84,90,96
What are the smallest numbers with exactly 13 factors, and
exactly 14 factors?
Answers: 4096 and 192.
How to calculate the probability that a factor has a certain
trait (even, primary, perfect square, etc…)?
Take a number (say 600). How many factors does it have?
200=3X5X5X2X2X2.
Thus factors are (3^0 or 3^1)X (5^0 or 5^1 or 5^2)X(2^0 or
2^1 or 2^2 or 2^3)
Total thus 2X3X4=24.
Modulo operations:
Operations of relatively primes.
What happens if the two numbers have common factors?
What can be told about powers of numbers?
Example: investigate 21 to the power 2015 (MOD 17).
Sets and subsets:
Definitions.
How to calculate the number of subsets?
What if there are other constraints on the number of subsets
(certain element, certain total number of elements, etc…)?
Practical probability questions.
How to calculate number of ways to seat around a round
table?
How to calculate constraints such as two individuals seating
next to each other?
What does it mean at random?
What does it mean probability, and how to define
conditional probability?
Example; you toss 4 coins and you are told that at least one
landed on head. What is the probability that exactly two
landed on head?