ISEE Lower 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 Lower 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: Identify whether $231$ is divisible by $3$.

Answer: Yes, $231$ is divisible by $3$. Sum of digits: 2 + 3 + 1 = 6, which is divisible by 3, so 231 is divisible by 3.

Flashcard 2: What is the prime factorization of $60$?

Answer: $2^2 \cdot 3 \cdot 5$. Decompose 60 as 2 × 30, then 2 × 2 × 15, and 3 × 5, yielding its prime factors.

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

Answer: Divisible by $3$ if the sum of digits is divisible by $3$. This stems from the fact that a number is congruent to the sum of its digits modulo 3.

Flashcard 4: What is the divisibility rule for $2$?

Answer: A number is divisible by $2$ if its last digit is even. This rule holds because in base 10, the number's evenness depends solely on its units digit being even.

Flashcard 5: What is the least common multiple (LCM) of two positive integers?

Answer: The smallest positive integer that is a multiple of both integers. This is the minimal number encompassing all prime factors of both at their highest powers.

Flashcard 6: What is the greatest common factor (GCF) of two integers?

Answer: The greatest positive integer that divides both integers. This measures the largest shared divisor, found via prime factorization or Euclidean algorithm.

Flashcard 7: What is a composite number?

Answer: A whole number greater than $1$ with more than two positive factors. This distinguishes numbers that are not prime, having factors beyond 1 and themselves.

Flashcard 8: What is a prime number?

Answer: A whole number greater than $1$ with exactly two positive factors. This definition identifies numbers only divisible by 1 and themselves, with no other factors.

Flashcard 9: What is the remainder when $53$ is divided by $5$?

Answer: $3$. 53 ÷ 5 = 10 with remainder 3, as 5 × 10 = 50 and 53 - 50 = 3.

Flashcard 10: Identify whether $1{,}248$ is divisible by $4$.

Answer: Yes, $1{,}248$ is divisible by $4$. Last two digits: 48 ÷ 4 = 12 (integer), confirming divisibility by 4.

Flashcard 11: What is the divisibility rule for $9$?

Answer: Divisible by $9$ if the sum of digits is divisible by $9$. This is due to the number being congruent to the sum of its digits modulo 9.

Flashcard 12: What is the divisibility rule for $10$?

Answer: Divisible by $10$ if the last digit is $0$. This rule holds because multiples of 10 in base 10 must end in 0.

Flashcard 13: What is the smallest positive integer divisible by both $9$ and $12$?

Answer: $36$. This is the LCM of 9 (3^2) and 12 (2^2 × 3), taking max exponents: 2^2 × 3^2 = 36.

Flashcard 14: What is the greatest prime factor of $45$?

Answer: $5$. Factorize 45 = 3^2 × 5; the prime factors are 3 and 5, with 5 being the largest.

Flashcard 15: Identify the greatest integer that divides both $48$ and $72$.

Answer: $24$. This is the GCF of 48 (2^4 × 3) and 72 (2^3 × 3^2), using min exponents: 2^3 × 3 = 24.

Flashcard 16: What is the divisibility rule for $4$?

Answer: Divisible by $4$ if the last $2$ digits form a multiple of $4$. This works as the last two digits represent the number modulo 100, and divisibility by 4 checks modulo 4.

Flashcard 17: What is the divisibility rule for $5$?

Answer: Divisible by $5$ if the last digit is $0$ or $5$. This rule applies because multiples of 5 in base 10 end in 0 or 5.

Flashcard 18: Identify whether $7{,}152$ is divisible by $8$.

Answer: Yes, $7{,}152$ is divisible by $8$. Last three digits: 152 ÷ 8 = 19 (integer), confirming divisibility by 8.

Flashcard 19: 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 both ensures divisibility by their product.

Flashcard 20: What is the divisibility rule for $8$?

Answer: Divisible by $8$ if the last $3$ digits form a multiple of $8$. This checks the number modulo 1000, as 1000 is divisible by 8, confirming divisibility by 8.

Flashcard 21: Which option is a factor of $84$: $5$, $6$, or $11$?

Answer: $6$. 84 ÷ 6 = 14 (integer), while 84 ÷ 5 = 16.8 and 84 ÷ 11 ≈ 7.636 are not integers.

Flashcard 22: Identify the LCM of $6$ and $8$.

Answer: $24$. Prime factors: 6 = 2 × 3, 8 = 2^3; LCM takes maximum exponents: 2^3 × 3 = 24.

Flashcard 23: Identify the GCF of $24$ and $36$.

Answer: $12$. Prime factors: 24 = 2^3 × 3, 36 = 2^2 × 3^2; GCF takes minimum exponents: 2^2 × 3 = 12.