ISEE Middle Level Quantitative Reasoning Flashcards: Divisibility And Factors

What this deck covers

This deck focuses on Divisibility And Factors, giving you a quick way to review the definitions, rules, and examples that matter most for ISEE Middle Level Quantitative Reasoning.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the definition of a prime number?

Answer: A number $>1$ with exactly two positive factors: $1$ and itself. Primes have no positive divisors other than 1 and themselves, distinguishing them from 1 and composites.

Flashcard 2: What is the definition of a composite number?

Answer: A number $>1$ with more than two positive factors. Composites have factors beyond 1 and themselves, unlike primes or 1.

Flashcard 3: What is the greatest common factor (GCF) of $18$ and $24$?

Answer: $6$. Prime factors: 18=2×3², 24=2³×3, so GCF=2×3=6.

Flashcard 4: What is the least common multiple (LCM) of $6$ and $15$?

Answer: $30$. Prime factors: 6=2×3, 15=3×5, so LCM=2×3×5=30.

Flashcard 5: Which statement is always true: If a number is divisible by $9$, is it divisible by $3$?

Answer: Yes, divisibility by $9$ always implies divisibility by $3$. Since 9=3², any multiple of 9 is also a multiple of 3.

Flashcard 6: Identify the missing digit $x$ so that $45x$ is divisible by $9$.

Answer: $x=0$ or $x=9$. Sum of digits 4+5+x=9+x must be divisible by 9, so for x=0-9, 9 or 18, thus x=0 or 9.

Flashcard 7: What is the prime factorization of $84$?

Answer: $2^2 \cdot 3 \cdot 7$. 84=2×42=2×2×21=2²×3×7, the complete prime factorization.

Flashcard 8: What is the greatest possible number of positive factors of $p^3$ where $p$ is prime?

Answer: $4$. For prime p, p³ has factors 1, p, p², p³, totaling 4, the maximum for that form.

Flashcard 9: What is the divisibility rule for $2$?

Answer: A number is divisible by $2$ if its last digit is even. A number's parity is determined solely by its last digit, so if even, the whole number is even and divisible by 2.

Flashcard 10: What is the divisibility rule for $3$?

Answer: Divisible by $3$ if the sum of digits is divisible by $3$. A number is congruent to the sum of its digits modulo 3, so if the sum is divisible by 3, the number is too.

Flashcard 11: What is the divisibility rule for $4$?

Answer: Divisible by $4$ if the last two digits form a multiple of $4$. The last two digits represent the number modulo 100, and since 100 is divisible by 4, divisibility depends on those digits.

Flashcard 12: What is the divisibility rule for $6$?

Answer: Divisible by $6$ if divisible by both $2$ and $3$. Since 6=2×3 and 2 and 3 are coprime, divisibility by 6 requires satisfying both rules.

Flashcard 13: What is the divisibility rule for $8$?

Answer: Divisible by $8$ if the last three digits form a multiple of $8$. The last three digits form the number modulo 1000, and 1000=8×125, so divisibility by 8 depends on those digits.

Flashcard 14: What is the divisibility rule for $9$?

Answer: Divisible by $9$ if the sum of digits is divisible by $9$. A number is congruent to the sum of its digits modulo 9, so if the sum is divisible by 9, the number is too.

Flashcard 15: What is the divisibility rule for $11$?

Answer: Divisible by $11$ if the alternating digit sum is a multiple of $11$. The alternating sum is equivalent to the number modulo 11, since 10≡-1 mod 11, alternating powers of -1.

Flashcard 16: Identify whether $378$ is divisible by $3$.

Answer: Yes, $378$ is divisible by $3$. Sum of digits 3+7+8=18, and 18÷3=6, so divisible by 3.

Flashcard 17: Identify whether $7,452$ is divisible by $4$.

Answer: Yes, $7,452$ is divisible by $4$. Last two digits 52, and 52÷4=13, an integer, so divisible by 4.

Flashcard 18: Identify whether $9,135$ is divisible by $5$.

Answer: Yes, $9,135$ is divisible by $5$. Ends with 5, which satisfies the divisibility rule for 5.

Flashcard 19: Identify whether $2,514$ is divisible by $6$.

Answer: Yes, $2,514$ is divisible by $6$. Even (ends with 4) and sum 2+5+1+4=12 divisible by 3, so divisible by 6.

Flashcard 20: Identify whether $12,488$ is divisible by $8$.

Answer: Yes, $12,488$ is divisible by $8$. Last three digits 488, and 488÷8=61, an integer, so divisible by 8.

Flashcard 21: Identify whether $4,554$ is divisible by $11$.

Answer: Yes, $4,554$ is divisible by $11$. Alternating sum 4-5+5-4=0, and 0 is a multiple of 11, so divisible by 11.