Number System and Divisibility Rules Flashcards

A complete set of vocabulary flashcards covering real and imaginary numbers, prime number statistics, divisibility rules for integers, and algebraic divisibility properties.

Rational Numbers

Numbers that can be written in $p/q$ form, including integers and fractions.

Irrational Numbers

Numbers that cannot be written in $p/q$ form, such as $\sqrt{3}$, $\sqrt{2}$ ($1.414\dots$), and $\pi$.

Terminating Decimal

A decimal that ends after a finite number of digits, like $1.6$, $2.73$, or $0.64$.

Non-terminating Repeating Decimal

A decimal that does not end but repeats a pattern, such as $0.5555\dots = 0.\overline{5}$.

Non-terminating Non-repeating Decimal

Decimals that are irrational and neither end nor repeat, like $0.46789\dots$ or $17.298765432\dots$.

Prime Number

Numbers that have exactly two factors, such as $2, 3, 5, 7, 11$.

Composite Number

Numbers with more than two factors, such as $4, 6, 8, 9, 10, 12, 14$.

Prime numbers from 1 to 100

There are exactly $25$ prime numbers in this range.

Prime numbers from 1 to 50

There are exactly $15$ prime numbers in this range.

Prime numbers from 50 to 100

There are exactly $10$ prime numbers in this range.

Prime numbers from 1 to 200

There are exactly $46$ prime numbers in this range.

Prime numbers from 1 to 1000

There are exactly $168$ prime numbers in this range.

Largest 2-digit Prime Number

$$97$$

Smallest and Largest 3-digit Prime Numbers

Smallest is $101$ and Largest is $997$.

Smallest and Largest 4-digit Prime Numbers

Smallest is $1009$ and Largest is $9973$.

Smallest and Largest 5-digit Prime Numbers

Smallest is $10007$ and Largest is $99991$.

Triple Prime Set

The only set of three prime numbers with a gap of $2$ between adjacent primes: $\{3, 5, 7\}$.

Co-primes (Relatively Prime)

Two numbers with a Highest Common Factor ($HCF$) of $1$ and only $1$ as a common factor, such as $(2,5)$ or $(35,18)$. Any two consecutive natural numbers are co-primes.

Twin Primes

Prime numbers with a difference of exactly $2$, such as $(11, 13)$ or $(29, 31)$. There are three twin primes between 35 and 100: $(41,43)$, $(59,61)$, and $(71,73)$.

Even and Odd Product Rules

$\text{even} \times \text{even} = \text{even}$, $\text{odd} \times \text{odd} = \text{odd}$, and $\text{even} \times \text{odd} = \text{even}$.

Even and Odd Addition Rules

$\text{even} + \text{even} = \text{even}$, $\text{odd} + \text{odd} = \text{even}$, and $\text{even} + \text{odd} = \text{odd}$.

Divisibility Rule for 3

The sum of the digits of the number must be divisible by $3$.

Divisibility Rule for 4

The last two digits of the number must form a number divisible by $4$.

Divisibility Rule for 6

The number must be divisible by both $2$ and $3$.

Divisibility Rule for 7

Double the last digit and subtract it from the remaining number; the result must be divisible by $7$ (e.g., for $483$, $48 - (3 \times 2) = 42$).

Divisibility Rule for 8

The last three digits of the number must form a number divisible by $8$.

Divisibility Rule for 11

The alternating sum of the digits must be divisible by $11$ (e.g., $10,813 \rightarrow 3-1+8-0+1 = 11$).

Divisibility Rule for 12

The number must be divisible by both $3$ and $4$.

The only even prime number

$$2$$

Terminating Condition for Rational Numbers

A rational number has a terminating decimal expansion if the divisor has no prime factors other than $2$ or $5$.

Divisibility of $a^n + b^n$

Divisible by $a+b$ when $n$ is odd; not divisible by $a+b$ when $n$ is even.

Divisibility of $a^n - b^n$

Divisible by $(a+b)$ and $(a-b)$ when $n$ is even; divisible by $a-b$ when $n$ is odd.

Remainder Theorem Equation

$\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}$.