Prime and Composite Numbers - SSAT Upper Level Quantitative

Identify whether $61$ is prime or composite.

$61$ is prime. No divisibility by primes up to $\sqrt{61} \approx 7.8$ (2, 3, 5, 7).
Identify whether $21$ is prime or composite.

$21$ is composite. Factors as $3 \times 7$ , showing more than two positive divisors.
Identify whether $53$ is prime or composite.

$53$ is prime. No divisibility by primes up to $\sqrt{53} \approx 7.3$ (2, 3, 5, 7).
Identify whether $97$ is prime or composite.

$97$ is prime. No divisibility by primes up to $\sqrt{97} \approx 9.8$ (2, 3, 5, 7).
Identify whether $35$ is prime or composite.

$35$ is composite. Factors as $5 \times 7$ , indicating multiple divisors.
What is the key stopping point when testing whether $n$ is prime by trial division?

Test primes up to and including $\sqrt{n}$ . Any factor larger than $\sqrt{n}$ pairs with one smaller, so checking up to this suffices.
What is the definition of a prime number in terms of its positive divisors?

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.
What is the definition of a composite number in terms of its positive divisors?

A number $>1$ with more than two positive divisors. Such numbers can be factored into products of smaller integers greater than 1.
Which statement is true: If $n$ is composite, then it has a prime factor $\le \sqrt{n}$ ?

True: every composite $n$ has a prime factor $\le \sqrt{n}$ . The smallest prime factor of a composite must not exceed its square root.
Identify whether $77$ is prime or composite.

$77$ is composite. Factors as $7 \times 11$ , showing it is not prime.
Identify whether $29$ is prime or composite.

$29$ is prime. No divisibility by primes up to $\sqrt{29} \approx 5.4$ (2, 3, 5).
Identify whether $51$ is prime or composite.

$51$ is composite. Factors as $3 \times 17$ , with more than two divisors.
Identify whether $17$ is prime or composite.

$17$ is prime. No divisibility by primes up to $\sqrt{17} \approx 4.1$ (i.e., 2 or 3).
Which number is the only even prime number?

$2$ . All even numbers greater than this are divisible by it and thus composite.
What is the correct classification of $1$ : prime, composite, or neither?

$1$ is neither prime nor composite. It has only one positive divisor, failing the criteria for both prime and composite.
Identify whether $49$ is prime or composite.

$49$ is composite. Factors as $7 \times 7$ , proving it is not prime.
What is the correct classification of $2$ : prime or composite?

$2$ is prime. Its only positive divisors are 1 and itself, meeting the prime definition despite being even.
What is the divisibility rule that shows a number ending in $0$ or $5$ is composite (if $>5$ )?

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 $n$ .
Identify whether $121$ is prime or composite.

$121$ is composite. Factors as $11 \times 11$ , indicating composite nature.
Identify whether $37$ is prime or composite.

$37$ is prime. No divisibility by primes up to $\sqrt{37} \approx 6.1$ (2, 3, 5).
What divisibility test proves an integer is composite if the sum of its digits is divisible by $3$ ?

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.
Identify whether $57$ is prime or composite.

$57$ is composite. Factors as $3 \times 19$ , confirming composite status.
Identify whether $91$ is prime or composite.

$91$ is composite. Factors as $7 \times 13$ , with multiple divisors.
What quick test proves an integer $n>2$ is composite if it is even?

If $2\mid n$ , then $n$ is composite. Even divisibility by 2 for $n>2$ means it has divisors other than 1 and itself.
Identify whether $59$ is prime or composite.

$59$ is prime. No divisibility by primes up to $\sqrt{59} \approx 7.7$ (2, 3, 5, 7).

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