Divisibility Rules and Number Theory Lecture

Flashcards covering various divisibility rules, numerical examples, and prime number identification based on the lecture notes.

Divisibility by $11$ (Rule)

A number is divisible by $11$ if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is $0$ or divisible by $11$.

Divisibility by $11$ (Example: $2189$)

The sum of digits in odd positions minus the sum of digits in even positions is $(9 + 1) - (8 + 2) = 10 - 10 = 0$, which is divisible by $11$.

Divisibility by $12$

A number is divisible by $12$ if it is divisible by both $3$ (sum of digits divisible by $3$) and $4$ (last two digits divisible by $4$).

Divisibility by $9$

A number is divisible by $9$ if the sum of its digits is divisible by $9$, such as the number $144$ where $1 + 4 + 4 = 9$.

Divisibility Strategy for Remainders

To find which number divided by $9$ leaves a remainder of $2$, subtract $2$ from each option and identify which resulting number is divisible by $9$ (e.g., $29 - 2 = 27$).

Least Common Multiple (LCM) Application

The time when three people who meet every $5$, $3$, and $2$ days will meet again is the smallest number divisible by all three, which is $30$ days.

Non-Prime Number Example ($87$)

As noted in the transcript, $87$ is not a prime number because the sum of its digits is $15$, making it divisible by $3$.

Divisibility by $2$

A number is divisible by $2$ if its units digit is $0, 2, 4, 6,$ or $8$.

Divisibility by $3$

A number is divisible by $3$ if the sum of its digits is divisible by $3$, such as $7302$ where the sum is $12$.

Divisibility by $4$

A number is divisible by $4$ if its units and tens digits form a number that is divisible by $4$, such as in $12728$ or $19324$.

Divisibility by $5$

A number is divisible by $5$ if its units digit is either $0$ or $5$.

Divisibility by $6$

A number is divisible by $6$ if it meets the criteria for divisibility by both $2$ and $3$.

Divisibility by $7$ (Rule)

A number is divisible by $7$ if the result of $(2 \times \text{Units Digit}) - (\text{Remaining Number})$ is divisible by $7$, such as $119$ where $(9 \times 2) - 11 = 7$.