Prime and Composite Numbers - SSAT Middle Level: Quantitative

$221$ is composite. 221 is $13 \times 17$, both primes greater than 1, confirming composite nature.

$143$ is composite. 143 factors as $11 \times 13$, with multiple factors beyond 1 and itself.

$169$ is composite. 169 equals $13^2$, resulting in factors 1, 13, and 169, classifying it as composite.

$51$ is composite. 51 is divisible by 3 (digit sum 6) and equals $3 \times 17$, confirming more than two factors.

$997$ is prime. 997 has no divisors among primes up to $\sqrt{997} \approx 31.6$, verifying it as prime.

$49$ is composite. 49 equals $7 \times 7$, giving it factors 1, 7, and 49, which exceeds two.

$29$ is prime. 29 has no divisors other than 1 and itself, as checks up to $\sqrt{29} \approx 5.4$ (primes 2, 3, 5) show no factors.

$n$ has a factor $d$ with $2 \le d \le \sqrt{n}$. For composites, factors come in pairs where one is at most $\sqrt{n}$, ensuring efficient identification.

Test prime divisors up to and including $ sqrt{n}$. Any factor larger than $\sqrt{n}$ pairs with one smaller than $\sqrt{n}$, so checking up to $\sqrt{n}$ suffices to determine primality.

If the alternating digit sum is a multiple of $11$, it is divisible by $11$. The alternating sum reflects the number's value modulo 11, where multiples of 11 yield results like 0, 11, or -11.

It is divisible by $9$ if the sum of its digits is divisible by $9$. Similar to the rule for 3, this checks congruence modulo 9 via digit sum for divisibility by 9.

It is divisible by $3$ if the sum of its digits is divisible by $3$. The rule works because a number is congruent to its digit sum modulo 3, allowing quick checks for factors of 3.

If it is divisible by $5$, it is composite. Numbers ending in 0 or 5 are divisible by 5, so those greater than 5 have 5 as a factor, making them composite.

A whole number $> 1$ with exactly two positive factors: $1$ and itself. This definition ensures primes cannot be formed by multiplying two smaller positive integers other than 1, making them fundamental in number theory.

If it is divisible by $2$, it is composite. Even numbers greater than 2 have 2 as a factor besides 1 and themselves, confirming they are not prime.

A whole number $> 1$ with more than two positive factors. Composites can be factored into products of smaller integers greater than 1, distinguishing them from primes.

$1$ is neither prime nor composite. By definition, 1 has only one positive factor, so it fits neither prime (exactly two factors) nor composite (more than two) categories.

$2$. All even numbers greater than 2 are divisible by 2, making them composite, while 2 has only 1 and itself as factors.

$57$ is composite. 57 is divisible by 3 (digit sum 12) and equals $3 \times 19$, indicating composite status.

$59$ is prime. 59 has no prime factors up to $\sqrt{59} \approx 7.7$ (2, 3, 5, 7), verifying its primality.

$61$ is prime. 61 remains undivided by primes up to $\sqrt{61} \approx 7.8$ (2, 3, 5, 7), establishing it as prime.

$77$ is composite. 77 factors as $7 \times 11$, both greater than 1, proving it composite.

$91$ is composite. 91 is $7 \times 13$, with factors including 1, 7, 13, and 91, exceeding two.

$121$ is composite. 121 is $11^2$, yielding factors 1, 11, and 121, making it composite.