GRE Quantitative Flashcards: Integer Properties Divisibility

What this deck covers

This deck focuses on Integer Properties Divisibility, giving you a quick way to review the definitions, rules, and examples that matter most for GRE Quantitative.

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 basic divisibility test for $3$ in base $10$?

Answer: Divisible by $3$ iff the sum of digits is divisible by $3$. This rule works because $10 \equiv 1 \pmod{3}$, so the number modulo 3 equals the digit sum modulo 3.

Flashcard 2: What is the remainder when $2^{10}$ is divided by $3$?

Answer: $1$. $2^{10} = 1024$, and $1024 \div 3 = 341$ with remainder 1.

Flashcard 3: What is the basic divisibility test for $6$ in base $10$?

Answer: Divisible by $6$ iff divisible by $2$ and by $3$. Since 6=2*3 and they are coprime, divisibility requires both conditions.

Flashcard 4: What is the basic divisibility test for $9$ in base $10$?

Answer: Divisible by $9$ iff the sum of digits is divisible by $9$. This follows since $10 \equiv 1 \pmod{9}$, equating the number to its digit sum modulo 9.

Flashcard 5: What is the value of $\gcd(a,b)$ when $a$ and $b$ are relatively prime?

Answer: $\gcd(a,b)=1$. Relatively prime integers share no common divisors other than 1.

Flashcard 6: What is the definition of the greatest common divisor $\gcd(a,b)$ for integers $a$ and $b$ not both $0$?

Answer: The greatest positive integer dividing both $a$ and $b$. It identifies the largest positive integer that divides both without remainder.

Flashcard 7: What is the definition of the least common multiple $\mathrm{lcm}(a,b)$ for nonzero integers $a$ and $b$?

Answer: The least positive integer that is a multiple of both $a$ and $b$. It specifies the smallest positive integer divisible by both $a$ and $b$.

Flashcard 8: What is the remainder when $10^{50}+3$ is divided by $10$?

Answer: $3$. $10^{50}$ is divisible by 10, so adding 3 gives remainder 3.

Flashcard 9: Identify the result: If $a \mid b$ and $a \mid c$, what must be true about $a \mid (b-c)$?

Answer: $a \mid (b-c)$. Divisibility holds for differences of multiples.

Flashcard 10: What is $\mathrm{lcm}(a,0)$ for a nonzero integer $a$?

Answer: $\mathrm{lcm}(a,0)=0$. Using the gcd-lcm relation, it evaluates to 0 for nonzero $a$.

Flashcard 11: What is the basic divisibility test for $10$ in base $10$?

Answer: Divisible by $10$ iff the last digit is $0$. In base 10, ending in 0 means it's a multiple of 10.

Flashcard 12: What is the basic divisibility test for $5$ in base $10$?

Answer: Divisible by $5$ iff the last digit is $0$ or $5$. In base 10, these endings make the number a multiple of 5.

Flashcard 13: What is the relationship between $\gcd(a,b)$ and $\mathrm{lcm}(a,b)$ for nonzero integers $a$ and $b$?

Answer: $\gcd(a,b)\,\mathrm{lcm}(a,b)=|ab|$. This formula relates gcd and lcm through the absolute product of the integers.

Flashcard 14: What is $\gcd(a,0)$ for a nonzero integer $a$?

Answer: $\gcd(a,0)=|a|$. Every integer divides 0, so the gcd is the absolute value of $a$.

Flashcard 15: What is the remainder when $7^{100}$ is divided by $7$?

Answer: $0$. Any positive power of 7 is divisible by 7, leaving no remainder.

Flashcard 16: Identify the result: If $a \mid b$, what must be true about $a \mid (bk)$ for any integer $k$?

Answer: $a \mid (bk)$ for every integer $k$. Scaling a multiple by any integer $k$ retains divisibility by $a$.

Flashcard 17: Identify whether $357$ is divisible by $3$.

Answer: Yes, $357$ is divisible by $3$. Digit sum $3+5+7=15$ is divisible by 3, confirming divisibility.

Flashcard 18: What is the basic divisibility test for $4$ in base $10$?

Answer: Divisible by $4$ iff the last two digits form a multiple of $4$. The last two digits represent the number modulo 100, and divisibility by 4 checks modulo 4.

Flashcard 19: What does it mean to say that $a$ is divisible by $b$ (with $b \ne 0$) in the integers?

Answer: $b \mid a$ means $a=bk$ for some integer $k$. This defines divisibility where $a$ is an integer multiple of $b$.

Flashcard 20: What is the basic divisibility test for $2$ in base $10$?

Answer: An integer is divisible by $2$ iff its last digit is even. In base 10, even last digits ensure the number is even.

Flashcard 21: What is the basic divisibility test for $12$ in base $10$?

Answer: Divisible by $12$ iff divisible by $3$ and by $4$. 12=3*4 with coprime factors, so both divisibility tests must hold.

Flashcard 22: What is the basic divisibility test for $8$ in base $10$?

Answer: Divisible by $8$ iff the last three digits form a multiple of $8$. The last three digits form the number modulo 1000, sufficient for divisibility by 8.

Flashcard 23: Identify the result: If $a \mid b$ and $a \mid c$, what must be true about $a \mid (b+c)$?

Answer: $a \mid (b+c)$. Divisibility preserves under addition of multiples.