Least Common Multiple and Divisibility Rules

Least Common Multiple

The common multiples of two or more numbers extend infinitely. For example, among the common multiples of the numbers $6$ and $8$, additional shared multiples beyond the first few include $72$, $96$, and so forth.

The smallest positive integer that is a multiple of both numbers is designated as the least common multiple. In Albanian mathematical notation, this is abbreviated as sh.v.p. (shumëfishi më i vogël i përbashkët). For the numbers $6$ and $8$, the least common multiple is $24$:

$\text{sh.v.p.}(6, 8) = 24$

Divisibility Rules

When determining whether a specific integer, such as $584$, is divisible by another integer like $8$, direct division can be carried out to check for a zero remainder. However, specific and simplified criteria exist to quickly test and verify divisibility without performing full division.

The formal divisibility rules (Kriteret e plotpjesëtimit) for positive integers are detailed below:

• Divisibility by $2$: All numbers whose last digit is an even number—specifically $0$, $2$, $4$, $6$, or $8$—are divisible by $2$.

• Divisibility by $3$: All numbers whose sum of digits equals a multiple of $3$ are divisible by $3$.

• Divisibility by $4$: All numbers that end in two zeros ($00$), or whose last two digits form a two-digit number that is a multiple of $4$, are divisible by $4$.

• Divisibility by $5$: All numbers whose last digit is either $0$ or $5$ are divisible by $5$.

• Divisibility by $6$: All numbers that are divisible simultaneously by both $2$ and $3$ are divisible by $6$.

• Divisibility by $7$: There is no simple general rule for testing divisibility by $7$.

• Divisibility by $8$: All numbers that end in three zeros ($000$), or whose last three digits form a number that is a multiple of $8$, are divisible by $8$.

• Divisibility by $9$: All numbers whose sum of digits equals a multiple of $9$ are divisible by $9$.

• Divisibility by $10$: All numbers that end in the digit $0$ are divisible by $10$.

• Divisibility by $100$: All numbers whose last two digits are $00$ are divisible by $100$.