Divisibility and Factors - ISEE Upper Level Quantitative Reasoning

1. What is the meaning of the greatest common factor (GCF) of two integers?

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

2. What is the meaning of $a\mid b$ in divisibility notation?

   $a\mid b$ 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.

3. What is the divisibility rule for $8$?

   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.

4. Identify whether $7{,}248$ is divisible by $8$.

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

5. What is the divisibility rule for $9$?

   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.

6. What is the divisibility rule for $2$?

   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.

7. What is the divisibility rule for $10$?

   Divisible by $10$ iff the last digit is $0$. Ending in $0$ ensures divisibility by $10 = 2 \times 5$, combining the rules for $2$ and $5$.

8. What is the meaning of the least common multiple (LCM) of two integers?

   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.

9. What is the LCM of $12$ and $18$?

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

10. Identify whether $12{,}456$ is divisible by $4$.

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

11. What is the divisibility rule for $6$?

    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.

12. Identify whether $8{,}235$ is divisible by $5$.

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

13. Identify whether $3{,}150$ is divisible by $6$.

    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 $3$.

14. Identify whether $4{,}719$ is divisible by $3$.

    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 $3$.

15. What is the divisibility rule for $11$?

    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.

16. What is the number of positive factors of $72$?

    $12$. 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.

17. What is the divisibility rule for $3$?

    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.

18. What is the GCF of $36$ and $54$?

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

19. Identify whether $5{,}736$ is divisible by $9$.

    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 $9$.

20. What is the definition of a composite number?

    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 $1$.

21. What is the prime factorization of $84$?

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

22. What is the definition of a prime number?

    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.

23. What is the divisibility rule for $4$?

    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.

24. What is the divisibility rule for $5$?

    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.

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