Integer Properties and Divisibility - GRE Quantitative Reasoning
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What is the basic divisibility test for $3$ in base $10$?
What is the basic divisibility test for $3$ in base $10$?
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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.
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.
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What is the basic divisibility test for $9$ in base $10$?
What is the basic divisibility test for $9$ in base $10$?
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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.
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.
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What is the basic divisibility test for $4$ in base $10$?
What is the basic divisibility test for $4$ in base $10$?
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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.
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.
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What is the basic divisibility test for $8$ in base $10$?
What is the basic divisibility test for $8$ in base $10$?
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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.
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.
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What is the basic divisibility test for $6$ in base $10$?
What is the basic divisibility test for $6$ in base $10$?
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Divisible by $6$ iff divisible by $2$ and by $3$. Since 6=2*3 and they are coprime, divisibility requires both conditions.
Divisible by $6$ iff divisible by $2$ and by $3$. Since 6=2*3 and they are coprime, divisibility requires both conditions.
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What is the basic divisibility test for $12$ in base $10$?
What is the basic divisibility test for $12$ in base $10$?
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Divisible by $12$ iff divisible by $3$ and by $4$. 12=3*4 with coprime factors, so both divisibility tests must hold.
Divisible by $12$ iff divisible by $3$ and by $4$. 12=3*4 with coprime factors, so both divisibility tests must hold.
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What is the definition of the least common multiple $\mathrm{lcm}(a,b)$ for nonzero integers $a$ and $b$?
What is the definition of the least common multiple $\mathrm{lcm}(a,b)$ for nonzero integers $a$ and $b$?
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The least positive integer that is a multiple of both $a$ and $b$. It specifies the smallest positive integer divisible by both $a$ and $b$.
The least positive integer that is a multiple of both $a$ and $b$. It specifies the smallest positive integer divisible by both $a$ and $b$.
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Identify the result: If $a \mid b$ and $a \mid c$, what must be true about $a \mid (b+c)$?
Identify the result: If $a \mid b$ and $a \mid c$, what must be true about $a \mid (b+c)$?
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$a \mid (b+c)$. Divisibility preserves under addition of multiples.
$a \mid (b+c)$. Divisibility preserves under addition of multiples.
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Identify the result: If $a \mid b$ and $a \mid c$, what must be true about $a \mid (b-c)$?
Identify the result: If $a \mid b$ and $a \mid c$, what must be true about $a \mid (b-c)$?
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$a \mid (b-c)$. Divisibility holds for differences of multiples.
$a \mid (b-c)$. Divisibility holds for differences of multiples.
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Identify the result: If $a \mid b$, what must be true about $a \mid (bk)$ for any integer $k$?
Identify the result: If $a \mid b$, what must be true about $a \mid (bk)$ for any integer $k$?
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$a \mid (bk)$ for every integer $k$. Scaling a multiple by any integer $k$ retains divisibility by $a$.
$a \mid (bk)$ for every integer $k$. Scaling a multiple by any integer $k$ retains divisibility by $a$.
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What is the remainder when $7^{100}$ is divided by $7$?
What is the remainder when $7^{100}$ is divided by $7$?
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$0$. Any positive power of 7 is divisible by 7, leaving no remainder.
$0$. Any positive power of 7 is divisible by 7, leaving no remainder.
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What is the remainder when $10^{50}+3$ is divided by $10$?
What is the remainder when $10^{50}+3$ is divided by $10$?
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$3$. $10^{50}$ is divisible by 10, so adding 3 gives remainder 3.
$3$. $10^{50}$ is divisible by 10, so adding 3 gives remainder 3.
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What is the remainder when $2^{10}$ is divided by $3$?
What is the remainder when $2^{10}$ is divided by $3$?
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$1$. $2^{10} = 1024$, and $1024 \div 3 = 341$ with remainder 1.
$1$. $2^{10} = 1024$, and $1024 \div 3 = 341$ with remainder 1.
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What does it mean to say that $a$ is divisible by $b$ (with $b \ne 0$) in the integers?
What does it mean to say that $a$ is divisible by $b$ (with $b \ne 0$) in the integers?
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$b \mid a$ means $a=bk$ for some integer $k$. This defines divisibility where $a$ is an integer multiple of $b$.
$b \mid a$ means $a=bk$ for some integer $k$. This defines divisibility where $a$ is an integer multiple of $b$.
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What is the value of $\gcd(a,b)$ when $a$ and $b$ are relatively prime?
What is the value of $\gcd(a,b)$ when $a$ and $b$ are relatively prime?
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$\gcd(a,b)=1$. Relatively prime integers share no common divisors other than 1.
$\gcd(a,b)=1$. Relatively prime integers share no common divisors other than 1.
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What is the relationship between $\gcd(a,b)$ and $\mathrm{lcm}(a,b)$ for nonzero integers $a$ and $b$?
What is the relationship between $\gcd(a,b)$ and $\mathrm{lcm}(a,b)$ for nonzero integers $a$ and $b$?
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$\gcd(a,b),\mathrm{lcm}(a,b)=|ab|$. This formula relates gcd and lcm through the absolute product of the integers.
$\gcd(a,b),\mathrm{lcm}(a,b)=|ab|$. This formula relates gcd and lcm through the absolute product of the integers.
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Identify whether $357$ is divisible by $3$.
Identify whether $357$ is divisible by $3$.
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Yes, $357$ is divisible by $3$. Digit sum $3+5+7=15$ is divisible by 3, confirming divisibility.
Yes, $357$ is divisible by $3$. Digit sum $3+5+7=15$ is divisible by 3, confirming divisibility.
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What is $\gcd(a,0)$ for a nonzero integer $a$?
What is $\gcd(a,0)$ for a nonzero integer $a$?
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$\gcd(a,0)=|a|$. Every integer divides 0, so the gcd is the absolute value of $a$.
$\gcd(a,0)=|a|$. Every integer divides 0, so the gcd is the absolute value of $a$.
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What is $\mathrm{lcm}(a,0)$ for a nonzero integer $a$?
What is $\mathrm{lcm}(a,0)$ for a nonzero integer $a$?
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$\mathrm{lcm}(a,0)=0$. Using the gcd-lcm relation, it evaluates to 0 for nonzero $a$.
$\mathrm{lcm}(a,0)=0$. Using the gcd-lcm relation, it evaluates to 0 for nonzero $a$.
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What is the definition of the greatest common divisor $\gcd(a,b)$ for integers $a$ and $b$ not both $0$?
What is the definition of the greatest common divisor $\gcd(a,b)$ for integers $a$ and $b$ not both $0$?
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The greatest positive integer dividing both $a$ and $b$. It identifies the largest positive integer that divides both without remainder.
The greatest positive integer dividing both $a$ and $b$. It identifies the largest positive integer that divides both without remainder.
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What is the basic divisibility test for $2$ in base $10$?
What is the basic divisibility test for $2$ in base $10$?
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An integer is divisible by $2$ iff its last digit is even. In base 10, even last digits ensure the number is even.
An integer is divisible by $2$ iff its last digit is even. In base 10, even last digits ensure the number is even.
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What is the basic divisibility test for $5$ in base $10$?
What is the basic divisibility test for $5$ in base $10$?
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Divisible by $5$ iff the last digit is $0$ or $5$. In base 10, these endings make the number a multiple of 5.
Divisible by $5$ iff the last digit is $0$ or $5$. In base 10, these endings make the number a multiple of 5.
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What is the basic divisibility test for $10$ in base $10$?
What is the basic divisibility test for $10$ in base $10$?
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Divisible by $10$ iff the last digit is $0$. In base 10, ending in 0 means it's a multiple of 10.
Divisible by $10$ iff the last digit is $0$. In base 10, ending in 0 means it's a multiple of 10.
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