Number Properties Flashcards
An integer is divisible by 3 if…
The sum of digits is divisible by 3
An integer is divisible by 4 if…
Integer is divisible by 2 twice or if last 2 digits is divisible by 4
An integer is divisible by 6 if…
The integer is divisible by both 2 and 3
An integer is divisible by 8 if…
The integer is divisible by 2 three times or if last 3 digits are divisible by 8
An integer is divisible by 9 if…
The sum of digits is divisible by 9
__ factors, __ multiples
Fewer factors, more multiples
If you add or subtract multiples of N, what does that mean for the result of that equation?
The result of adding and subtracting a multiple of N is a multiple of N
What is the factor foundation rule?
If a is a factor of b, and b is a factor of c, then a is a factor of c.
The remainder is (larger/smaller) than the divisor?
Smaller
Dividend = __ x __ + ___
Dividend = Quotient x Divisor + Remainder
Even +/- Even =
Even
Odd +/- Odd =
Even
Even +/- Odd =
Odd
Even x Even =
Even
Even x Odd =
Even
Odd x Odd =
Odd
The sum of any two prime numbers is (odd / even)?
Even, unless one of the primes is 2, then the sum of 2 primes is odd
What are the steps for testing cases?
1) What possible cases are allowed?
2) Choose numbers that work for the statement.
3) Try to prove the statement is insufficient.
What is considered “insufficient” for DS questions?
When there are 2+ answers, or situations where the answer could be either yes or no
For PS problems, what does “must be” or “could be” mean?
Consider how to disprove the other 4 answers.
What are the categories to consider when picking numbers to test?
1) Even or odd
2) Fractions between 0 and -1
3) Fractions between 0 and 1
4) Primes
What types of numbers do you pick when testing for absolute values?
Negatives
What types of numbers do you use when testing for exponents?
0 or 1
What do you test for fractions?
Numbers between 0 and 1
OR means __ and AND means __
OR means add, AND means multiply
“How many” is a __ problem.
Combinatorics
A factorial is…
…the number of ways to arrange n distinct objects if no restrictions are stated.
If making different decisions, determine the concepts separately and __ the results.
Determine the ideas separately and multiply the results.
“At most” means that you have __ decisions.
You have multiple decisions and it is an OR decision
In probability, numerators and denominators are related but must be calculated (together/separately).
Separately
Equation for probability adding to 1
P(A) + P(not A) = 1
In probability, “at least” / “at most” means that you calculate the probability in what way?
Calculate the probability that something doesn’t happen, based on what the problem specifies.
What makes a number prime?
It has only 2 factors.
If you add/subtract a multiple of N with a non-multiple of N…
The result is a non-multiple of N.
If you add 2 non-multiples of N, the result…
…the result could either be a multiple or not.
What is the Greatest Common Factor?
It is the largest divisor of 2+ integers, and it smaller than or equal to the starting integers.
What is the Least Common Multiple?
It is the smallest multiple of 2+ integers, and the multiple will be larger than or equal to the starting integers.
If a prime factor appears to the Nth power, there are __ possibilities for occurrences of that prime factor.
There are N+1 possibilities.
If a number has a prime factorization of (a)(b)(c), all raised to different exponents (x, y, and z)…
…then the number has (x+1)(y+1)(z+1) different factors.
All perfect squares have (odd/even) number of total factors.
Odd.
The prime factorization of a perfect square has only (odd/even) powers of primes.
Even. Numbers with odd powers of primes are not perfect squares.
N! is a multiple of all integers from…
All integers from 1 to N.
Eg. 10! + 7 is a multiple of 7.
What is the glue method for combinatorics problems?
Where items or people must be next to each other, pretend all the items are “glued” together as a larger item.
Find the total possibilities then subtract the circumstances where it can’t work.
What is the domino effect rule?
The rule states that you should multiply probabilities of events in a sequence, taking earlier events into account.
How do you calculate a problem with a symmetrical situation, with multiple equivalent cases?
Calculate the probability of one case and multiply by the number of possible cases to find the probability of any of the cases.
