Prime and Composite Numbers - SSAT Upper Level: Quantitative
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Identify whether $77$ is prime or composite.
Identify whether $77$ is prime or composite.
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$77$ is composite. Factors as $7 \times 11$, showing it is not prime.
$77$ is composite. Factors as $7 \times 11$, showing it is not prime.
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Identify whether $121$ is prime or composite.
Identify whether $121$ is prime or composite.
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$121$ is composite. Factors as $11 \times 11$, indicating composite nature.
$121$ is composite. Factors as $11 \times 11$, indicating composite nature.
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Identify whether $51$ is prime or composite.
Identify whether $51$ is prime or composite.
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$51$ is composite. Factors as $3 \times 17$, with more than two divisors.
$51$ is composite. Factors as $3 \times 17$, with more than two divisors.
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Identify whether $57$ is prime or composite.
Identify whether $57$ is prime or composite.
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$57$ is composite. Factors as $3 \times 19$, confirming composite status.
$57$ is composite. Factors as $3 \times 19$, confirming composite status.
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Identify whether $59$ is prime or composite.
Identify whether $59$ is prime or composite.
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$59$ is prime. No divisibility by primes up to $\sqrt{59} \approx 7.7$ (2, 3, 5, 7).
$59$ is prime. No divisibility by primes up to $\sqrt{59} \approx 7.7$ (2, 3, 5, 7).
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Identify whether $29$ is prime or composite.
Identify whether $29$ is prime or composite.
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$29$ is prime. No divisibility by primes up to $\sqrt{29} \approx 5.4$ (2, 3, 5).
$29$ is prime. No divisibility by primes up to $\sqrt{29} \approx 5.4$ (2, 3, 5).
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Identify whether $97$ is prime or composite.
Identify whether $97$ is prime or composite.
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$97$ is prime. No divisibility by primes up to $\sqrt{97} \approx 9.8$ (2, 3, 5, 7).
$97$ is prime. No divisibility by primes up to $\sqrt{97} \approx 9.8$ (2, 3, 5, 7).
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Identify whether $53$ is prime or composite.
Identify whether $53$ is prime or composite.
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$53$ is prime. No divisibility by primes up to $\sqrt{53} \approx 7.3$ (2, 3, 5, 7).
$53$ is prime. No divisibility by primes up to $\sqrt{53} \approx 7.3$ (2, 3, 5, 7).
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Identify whether $91$ is prime or composite.
Identify whether $91$ is prime or composite.
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$91$ is composite. Factors as $7 \times 13$, with multiple divisors.
$91$ is composite. Factors as $7 \times 13$, with multiple divisors.
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Identify whether $61$ is prime or composite.
Identify whether $61$ is prime or composite.
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$61$ is prime. No divisibility by primes up to $\sqrt{61} \approx 7.8$ (2, 3, 5, 7).
$61$ is prime. No divisibility by primes up to $\sqrt{61} \approx 7.8$ (2, 3, 5, 7).
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Identify whether $49$ is prime or composite.
Identify whether $49$ is prime or composite.
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$49$ is composite. Factors as $7 \times 7$, proving it is not prime.
$49$ is composite. Factors as $7 \times 7$, proving it is not prime.
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Identify whether $37$ is prime or composite.
Identify whether $37$ is prime or composite.
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$37$ is prime. No divisibility by primes up to $\sqrt{37} \approx 6.1$ (2, 3, 5).
$37$ is prime. No divisibility by primes up to $\sqrt{37} \approx 6.1$ (2, 3, 5).
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Identify whether $35$ is prime or composite.
Identify whether $35$ is prime or composite.
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$35$ is composite. Factors as $5 \times 7$, indicating multiple divisors.
$35$ is composite. Factors as $5 \times 7$, indicating multiple divisors.
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Identify whether $21$ is prime or composite.
Identify whether $21$ is prime or composite.
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$21$ is composite. Factors as $3 \times 7$, showing more than two positive divisors.
$21$ is composite. Factors as $3 \times 7$, showing more than two positive divisors.
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Identify whether $17$ is prime or composite.
Identify whether $17$ is prime or composite.
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$17$ is prime. No divisibility by primes up to $\sqrt{17} \approx 4.1$ (i.e., 2 or 3).
$17$ is prime. No divisibility by primes up to $\sqrt{17} \approx 4.1$ (i.e., 2 or 3).
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Which statement is true: If $n$ is composite, then it has a prime factor $\le \sqrt{n}$?
Which statement is true: If $n$ is composite, then it has a prime factor $\le \sqrt{n}$?
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True: every composite $n$ has a prime factor $\le \sqrt{n}$. The smallest prime factor of a composite must not exceed its square root.
True: every composite $n$ has a prime factor $\le \sqrt{n}$. The smallest prime factor of a composite must not exceed its square root.
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What is the key stopping point when testing whether $n$ is prime by trial division?
What is the key stopping point when testing whether $n$ is prime by trial division?
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Test primes up to and including $\sqrt{n}$. Any factor larger than $\sqrt{n}$ pairs with one smaller, so checking up to this suffices.
Test primes up to and including $\sqrt{n}$. Any factor larger than $\sqrt{n}$ pairs with one smaller, so checking up to this suffices.
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What is the divisibility rule that shows a number ending in $0$ or $5$ is composite (if $>5$)?
What is the divisibility rule that shows a number ending in $0$ or $5$ is composite (if $>5$)?
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If last digit is $0$ or $5$, then $5\mid n$ so composite for $n>5$. Ending in 0 or 5 indicates divisibility by 5, confirming factors other than 1 and $n$.
If last digit is $0$ or $5$, then $5\mid n$ so composite for $n>5$. Ending in 0 or 5 indicates divisibility by 5, confirming factors other than 1 and $n$.
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What divisibility test proves an integer is composite if the sum of its digits is divisible by $3$?
What divisibility test proves an integer is composite if the sum of its digits is divisible by $3$?
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If $3\mid$ (digit sum), then $3\mid n$ so $n$ is composite. Divisibility by 3 for $n>3$ implies additional factors beyond 1 and itself.
If $3\mid$ (digit sum), then $3\mid n$ so $n$ is composite. Divisibility by 3 for $n>3$ implies additional factors beyond 1 and itself.
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What quick test proves an integer $n>2$ is composite if it is even?
What quick test proves an integer $n>2$ is composite if it is even?
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If $2\mid n$, then $n$ is composite. Even divisibility by 2 for $n>2$ means it has divisors other than 1 and itself.
If $2\mid n$, then $n$ is composite. Even divisibility by 2 for $n>2$ means it has divisors other than 1 and itself.
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Which number is the only even prime number?
Which number is the only even prime number?
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$2$. All even numbers greater than this are divisible by it and thus composite.
$2$. All even numbers greater than this are divisible by it and thus composite.
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What is the correct classification of $2$: prime or composite?
What is the correct classification of $2$: prime or composite?
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$2$ is prime. Its only positive divisors are 1 and itself, meeting the prime definition despite being even.
$2$ is prime. Its only positive divisors are 1 and itself, meeting the prime definition despite being even.
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What is the correct classification of $1$: prime, composite, or neither?
What is the correct classification of $1$: prime, composite, or neither?
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$1$ is neither prime nor composite. It has only one positive divisor, failing the criteria for both prime and composite.
$1$ is neither prime nor composite. It has only one positive divisor, failing the criteria for both prime and composite.
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What is the definition of a composite number in terms of its positive divisors?
What is the definition of a composite number in terms of its positive divisors?
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A number $>1$ with more than two positive divisors. Such numbers can be factored into products of smaller integers greater than 1.
A number $>1$ with more than two positive divisors. Such numbers can be factored into products of smaller integers greater than 1.
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What is the definition of a prime number in terms of its positive divisors?
What is the definition of a prime number in terms of its positive divisors?
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A number $>1$ with exactly two positive divisors: $1$ and itself. This criterion ensures the number cannot be expressed as a product of two integers greater than 1.
A number $>1$ with exactly two positive divisors: $1$ and itself. This criterion ensures the number cannot be expressed as a product of two integers greater than 1.
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