Factors, Divisibility, Primes, and Factorials - Vocabulary Flashcards

Vocabulary flashcards covering factors, divisibility, primes, and factorials concepts from the lecture.

Factor

A number that divides another number exactly (the top divided by the bottom yields an integer).

Divisor

Another term for a factor; the number you divide by when checking divisibility.

Factor pair

Two numbers a and b such that a × b equals a given number; both divide that number.

Positive factors

Factors that are positive integers.

Negative factors

Negative counterparts of positive factors.

Prime number

A positive integer greater than 1 with exactly two distinct positive factors: 1 and itself.

Composite number

A positive integer greater than 1 that has more than two positive factors.

Prime factorization

Expressing a number as a product of primes raised to powers; unique by the Fundamental Theorem of Arithmetic.

Greatest common factor (GCF)

The largest positive integer that divides two or more numbers exactly.

Least common multiple (LCM)

The smallest positive integer that is a multiple of two or more numbers.

Factorial

n! = n × (n−1) × … × 2 × 1; by convention 0! = 1.

Trailing zeros

Number of zeros at the end of n!; determined by the number of factor 10s, i.e., pairs of 2s and 5s.

Unit digit

The last digit of an integer; used to determine unit digits of powers.

Unit digit pattern for 2^n

Cycle of unit digits: 2, 4, 8, 6, repeating every 4.

Unit digit pattern for 3^n

Cycle of unit digits: 3, 9, 7, 1, repeating every 4.

Unit digit pattern for 4^n

Cycle: 4, 6, 4, 6, repeating every 2.

Unit digit pattern for 5^n

Unit digit is always 5 (for n ≥ 1).

Unit digit pattern for 6^n

Unit digit is always 6 (for n ≥ 1).

Divisibility rule for 2

A number is divisible by 2 if its unit digit is even (0,2,4,6,8).

Divisibility rule for 3

A number is divisible by 3 if the sum of its digits is a multiple of 3.

Divisibility rule for 4

A number is divisible by 4 if its last two digits form a multiple of 4.

Divisibility rule for 5

A number is divisible by 5 if it ends in 0 or 5.

Divisibility rule for 6

Divisible by 6 if it is divisible by both 2 and 3.

Divisibility rule for 8

A number is divisible by 8 if its last three digits form a multiple of 8.

Divisibility rule for 9

A number is divisible by 9 if the sum of its digits is a multiple of 9.

Divisibility rule for 10

A number is divisible by 10 if it ends in 0.

Divisibility rule for 11

A number is divisible by 11 if the alternating sum of its digits (left to right) is 0 or a multiple of 11.

Three consecutive integers

Three integers in a row: n, n+1, n+2 (or any permutation).

Sum of three consecutive integers

The sum is always a multiple of 3; it is even if the first is odd, and odd if the first is even.

Product of three consecutive integers

The product is always a multiple of 6; in some cases it is a multiple of 8 if the middle term is odd.

Even vs. odd three-term sequences

Two cases: even, odd, even or odd, even, odd.

Euclid’s proof of infinite primes

Assume finitely many primes; multiply them and add 1 to get a new prime or new prime factor, contradicting finiteness.

Prime factorization as DNA

Prime factors uniquely describe a number; e.g., 100 = 2^2 × 5^2.

Number of positive factors from prime factorization

If n = ∏ pi^{ai}, then number of positive factors = ∏ (a_i+1).

Odd positive factors

Count factors using only odd primes (ignore the factor 2).

Even positive factors

Total positive factors minus the odd positive factors.

Interval multiples formula (count)

Number of multiples of k in [A,B] is ⌊B/k⌋ − ⌊(A−1)/k⌋.

Sum of multiples in an interval (pattern)

Sum can be found by pairing first and last multiples and using arithmetic series.

Three consecutive integers (algebra form)

Represented as n, n+1, n+2 or x−1, x, x+1 to handle algebra problems.

Zero

Zero is even, not prime, a factor of only itself; 0! = 1; 0^0 is undefined; 0 is a multiple of every integer.

Quotient and remainder

In division x ÷ y, quotient is how many times y fits; remainder is what’s left and is nonnegative.

Prime testing up to sqrt(n)

To test primality, only check divisibility by primes up to √n.

Euclid’s primes and intervals

Primes become sparser at large numbers, but there are infinitely many primes.

Trailing zeros calculation example

Count factors of 5 in n! (plus 5^2, 5^3, …) to determine zeros at the end.

Factor

A number that divides another number exactly (the top divided by the bottom yields an integer).

Divisor

Another term for a factor; the number you divide by when checking divisibility.

Factor pair

Two numbers a and b such that a × b equals a given number; both divide that number.

Positive factors

Factors that are positive integers.

Negative factors

Negative counterparts of positive factors.

Prime number

A positive integer greater than 1 with exactly two distinct positive factors: 1 and itself.

Composite number

A positive integer greater than 1 that has more than two positive factors.

Prime factorization

Expressing a number as a product of primes raised to powers; unique by the Fundamental Theorem of Arithmetic.

Greatest common factor (GCF)

The largest positive integer that divides two or more numbers exactly.

Least common multiple (LCM)

The smallest positive integer that is a multiple of two or more numbers.

Factorial

n! = n × (n−1) × … × 2 × 1; by convention 0! = 1.

Trailing zeros

Number of zeros at the end of n!; determined by the number of factor 10s, i.e., pairs of 2s and 5s.

Unit digit

The last digit of an integer; used to determine unit digits of powers.

Unit digit pattern for 2^n

Cycle of unit digits: 2, 4, 8, 6, repeating every 4.

Unit digit pattern for 3^n

Cycle of unit digits: 3, 9, 7, 1, repeating every 4.

Unit digit pattern for 4^n

Cycle: 4, 6, 4, 6, repeating every 2.

Unit digit pattern for 5^n

Unit digit is always 5 (for n ≥ 1).

Unit digit pattern for 6^n

Unit digit is always 6 (for n ≥ 1).

Divisibility rule for 2

A number is divisible by 2 if its unit digit is even (0,2,4,6,8).

Divisibility rule for 3

A number is divisible by 3 if the sum of its digits is a multiple of 3.

Divisibility rule for 4

A number is divisible by 4 if its last two digits form a multiple of 4.

Divisibility rule for 5

A number is divisible by 5 if it ends in 0 or 5.

Divisibility rule for 6

Divisible by 6 if it is divisible by both 2 and 3.

Divisibility rule for 8

A number is divisible by 8 if its last three digits form a multiple of 8.

Divisibility rule for 9

A number is divisible by 9 if the sum of its digits is a multiple of 9.

Divisibility rule for 10

A number is divisible by 10 if it ends in 0.

Divisibility rule for 11

A number is divisible by 11 if the alternating sum of its digits (left to right) is 0 or a multiple of 11.

Three consecutive integers

Three integers in a row: n, n+1, n+2 (or any permutation).

Sum of three consecutive integers

The sum is always a multiple of 3; it is even if the first is odd, and odd if the first is even.

Product of three consecutive integers

The product is always a multiple of 6; in some cases it is a multiple of 8 if the middle term is odd.

Even vs. odd three-term sequences

Two cases: even, odd, even or odd, even, odd.

Euclid’s proof of infinite primes

Assume finitely many primes; multiply them and add 1 to get a new prime or new prime factor, contradicting finiteness.

Prime factorization as DNA

Prime factors uniquely describe a number; e.g., 100 = 2^2 × 5^2.

Number of positive factors from prime factorization

If n = ∏ pi^{ai}, then number of positive factors = ∏ (a_i+1).

Odd positive factors

Count factors using only odd primes (ignore the factor 2).

Even positive factors

Total positive factors minus the odd positive factors.

Interval multiples formula (count)

Number of multiples of k in [A,B] is ⌊B/k⌋ − ⌊(A−1)/k⌋.

Sum of multiples in an interval (pattern)

Sum can be found by pairing first and last multiples and using arithmetic series.

Three consecutive integers (algebra form)

Represented as n, n+1, n+2 or x−1, x, x+1 to handle algebra problems.

Zero

Zero is even, not prime, a factor of only itself; 0! = 1; 0^0 is undefined; 0 is a multiple of every integer.

Quotient and remainder

In division x ÷ y, quotient is how many times y fits; remainder is what’s left and is nonnegative.

Prime testing up to sqrt(n)

To test primality, only check divisibility by primes up to √n.

Euclid’s primes and intervals

Primes become sparser at large numbers, but there are infinitely many primes.

Trailing zeros calculation example

Count factors of 5 in n! (plus 5^2, 5^3, …) to determine zeros at the end.

Relatively Prime Numbers (Coprime)

Two integers are relatively prime (or coprime) if their greatest common factor (GCF) is 1.

Multiple

A multiple of a number is the product of that number and any integer.

Perfect Number

A positive integer that is equal to the sum of its positive divisors, excluding the number itself.

Factor

A number that divides another number exactly (the top divided by the bottom yields an integer).

Divisor

Another term for a factor; the number you divide by when checking divisibility.

Factor pair

Two numbers a and b such that a × b equals a given number; both divide that number.

Positive factors

Factors that are positive integers.

Negative factors

Negative counterparts of positive factors.

Prime number

A positive integer greater than 1 with exactly two distinct positive factors: 1 and itself.

Composite number

A positive integer greater than 1 that has more than two positive factors.

Prime factorization

Expressing a number as a product of primes raised to powers; unique by the Fundamental Theorem of Arithmetic.

Greatest common factor (GCF)

The largest positive integer that divides two or more numbers exactly.