Properties of Numbers Flashcards
(19 cards)
6 Properties of Zero
- x/0 is undefined
- square root (0) = 0
- NEITHER positive nor negative
- The only number that is EQUAL TO ITS OPPOSITE
- All numbers are factors of 0 and 0 is a multiple of all numbers
- Zero is an EVEN number
Factors
Given 2 whole numbers, X is a factor of Y if X divides evenly (int.) into Y.
First 25 prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Finding Total # of Factors by Prime Factorization
- Find the Prime Factorization
- Add 1 to the value of each exponent & multiply results
Finding Total # of ODD Factors by Prime Factorization
- Find the Prime Factorization
- Remove the power w/ Base 2
- Add 1 to each of the remaining exponents & multiple results
Finding Least Common Multiple (LCM) from Prime Factorization
- Find the Prime Factorization
- Of any REPEATING prime factors, take only those w/ the LARGEST exponent
- Of what’s left, take all non-repeated exponents
- Multiply together. Result = LCM
Finding Greatest Common Factor (GCF) from Prime Factorization
- Find the Prime Factorization
- Identity REPEATING Prime Factors
- Of repeating prime factors, take only those with the SMALLEST exponent
- Multiply together. Result = GCF
Relationship between GCF and LCM
If we know the LCM & GCF of 2 positive integers, we know the product!
Product = LCM x GCF
(X*Y) = LCM x GCF
What Formula Can We Use for Repeating Pattern Questions (“shared original starting point”)?
LCM!
Prime Factorization and Divisibility
Positive int. X is divisible by positive int. Y ONLY IF the prime factorization of X contains the prime factorization of Y.
j, k, and N relationship = jN/k
j/k is FULLY REDUCED.
Then, N must be divisible by K!
Ex: 5N/4 == N must be divisible by 4
Divisibility rules of the Product of Consecutive Integers & Patterns to watch for
The product of any N # consecutive integers is ALWAYS divisible by N!
1) n^2 - n = N(n-1) = 2 consecutive
2) n^2 + n = N(n+1) = 2 consecutive
3) n^3 - n = (n-1)(n)(n+1) = 3 consecutive
4) n^5-5n^3 +4n = (n)(n+1)(n-1)(n+2)(n-2) = 5 consecutive
Combining Remainder Information
If:
positive int. N divided by positive int. J leaves remainder of B
&
positive int. N divided by positive int. K leaves remainder C
- Find the smallest possible value of N [satisfied both remainders – first common]
- Add the LCM of J and K to the smallest value as many times as necessary
Determining the Number of TRAILING Zeros in a Number
In whole numbers, trailing zeros are created by (2 x 5) pairs. Each (2 x 5) pair creates one trailing zero.
Any factorial greater than or equal to 5! will have a trailing zero.
10 = 1
100 = 2
1000 = 3
Determining the Number of LEADING Zeros in a Number
When the denominator is NOT a perfect power of 10:
If X is an int. with K number of digits, 1/x will have k-1 leading zeros
When the denominator IS a perfect power of 10: k-2 leading zeros
Shortcut for Determining # of Primes in a Factor
Ex: 21!/3^n
- Divide 21 by 3^1, 3^2, 3^k
[Ignore any remainders] - Add up all of the quotients from the division. The quotient represents the # of 3s in the prime factorization of 21!
Easy way to determine if there is a terminating decimal
A fraction/decimal will only terminate IF AND ONLY OF the denominator of the reduced fraction has a prime factorization that contains ONLY 2s and 5s, or both.
Patterns in Units Digits
0: 0
1: 1
2: 2-4-6-8
3: 3-9-7-1
4: 4-6 [all positive: 4; all negative: 6]
5: 5
6: 6
7: 7-9-3-1
8: 8-4-2-6
9: 9-1 [all positive: 1; all negative: 9]
Remainders after Division by 5
Integers with the same units digit when divided by 5 will ALWAYS have the same remainder
