Divisibility and Factors - ISEE Upper Level: Quantitative Reasoning

Divisible by $11$ iff the alternating digit sum is a multiple of $11$. The alternating sum is congruent to the number modulo $11$ because $10 \equiv -1 \pmod{11}$, alternating signs in place values.

Yes; $248$ is divisible by $8$. The last three digits, $248$, divided by $8$ yield $31$, confirming divisibility as per the rule for $8$.

No; $5+7+3+6=21$ is not divisible by $9$. The digit sum $21$ is not a multiple of $9$ ($21 \div 9 = 2.333$), so the number fails the divisibility rule for $9$.

Yes; it is divisible by $2$ and $3$. The number is even and its digit sum $9$ is divisible by $3$, satisfying the combined rules for $2$ and $3$.

Yes; the last digit is $5$. Ending in $5$ directly applies the divisibility rule for $5$ in base-10 numerals.

Yes; $56$ is divisible by $4$. The last two digits, $56$, divided by $4$ yield $14$, meeting the criterion for divisibility by $4$.

Yes; $4+7+1+9=21$ is divisible by $3$. The digit sum $21$ is divisible by $3$ ($21 \div 3 = 7$), adhering to the rule for divisibility by $3$.

An integer $>1$ with exactly two positive factors: $1$ and itself. Primes are defined this way to distinguish numbers with no divisors other than $1$ and themselves, ensuring they are building blocks of integers.

An integer $>1$ with more than two positive factors. Composites have factors beyond $1$ and themselves, distinguishing them from primes and $1$ in the classification of integers greater than $1$.

The greatest positive integer that divides both integers. The GCF represents the largest shared divisor, useful for simplifying fractions and solving Diophantine equations.

The smallest positive integer that is a multiple of both integers. The LCM is the smallest common multiple, essential for adding fractions and finding periodic alignments in number theory.

$84=2^2\cdot 3\cdot 7$. Dividing $84$ repeatedly by smallest primes yields $2^2 \times 3 \times 7$, capturing its unique prime composition under the fundamental theorem of arithmetic.

$18$. Using prime factorizations ($36=2^2 \times 3^2$, $54=2 \times 3^3$), the GCF takes minimum exponents: $2^1 \times 3^2 = 18$.

$36$. From prime factorizations ($12=2^2 \times 3$, $18=2 \times 3^2$), the LCM uses maximum exponents: $2^2 \times 3^2 = 36$.

$12$. Prime factorization $72=2^3 \times 3^2$ gives number of factors as $(3+1)(2+1)=12$, by adding one to each exponent and multiplying.

$a\mid b$ means $b=ak$ for some integer $k$. This notation indicates $a$ divides $b$ exactly, meaning $b$ is an integer multiple of $a$, fundamental in number theory.

A number is divisible by $2$ iff its last digit is even. This rule identifies even numbers, which are multiples of $2$ due to the base-10 system's parity preservation in the units place.

Divisible by $6$ iff divisible by both $2$ and $3$. Since $6 = 2 \times 3$ and $2$ and $3$ are coprime, a number must satisfy both divisibility rules simultaneously.

Divisible by $5$ iff the last digit is $0$ or $5$. In base-10, numbers ending in $0$ or $5$ are multiples of $5$ as they align with the decimal place value system's structure.

Divisible by $8$ iff the last three digits form a multiple of $8$. The last three digits form the number modulo $1000$, and $1000 = 8 \times 125$, so divisibility by $8$ is determined by this segment.

Divisible by $4$ iff the last two digits form a multiple of $4$. The last two digits represent the number modulo $100$, and since $100 = 4 \times 25$, divisibility by $4$ depends on this portion.

Divisible by $3$ iff the sum of digits is divisible by $3$. The sum of digits is congruent to the number modulo $3$ because $10 \equiv 1 \pmod{3}$, making the rule effective for checking divisibility.

Divisible by $9$ iff the sum of digits is divisible by $9$. The sum of digits is congruent to the number modulo $9$ because $10 \equiv 1 \pmod{9}$, allowing efficient divisibility checking.