Prime and Composite Numbers - SSAT Middle Level Quantitative

1. What is the only even prime number?

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

2. What quick test shows a whole number $> 5$ is composite if it ends in $0$ or $5$?

   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.

3. Identify whether $997$ is prime or composite.

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

4. What is the definition of a composite number?

   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.

5. What is the stopping rule for testing whether $n$ is prime using trial division?

   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.

6. Which statement is always true for a composite number $n$?

   $n$ 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.

7. Identify whether $57$ is prime or composite.

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

8. What divisibility rule tests whether a number is divisible by $9$?

   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.

9. What divisibility rule tests whether a number is divisible by $11$?

   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.

10. Identify whether $91$ is prime or composite.

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

11. Identify whether $29$ is prime or composite.

    $29$ 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.

12. Identify whether $61$ is prime or composite.

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

13. Identify whether $59$ is prime or composite.

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

14. Identify whether $51$ is prime or composite.

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

15. Identify whether $169$ is prime or composite.

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

16. Identify whether $97$ is prime or composite.

    $97$ is prime. 97 is not divisible by primes up to $\sqrt{97} \approx 9.8$ (2, 3, 5, 7), confirming primality.

17. What is the definition of a prime number?

    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.

18. What divisibility rule tests whether a number is divisible by $3$?

    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.

19. What is the correct classification of the number $1$: prime, composite, or neither?

    $1$ 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.

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

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

21. Identify whether $77$ is prime or composite.

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

22. Identify whether $221$ is prime or composite.

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

23. Identify whether $143$ is prime or composite.

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

24. What quick test shows a whole number $> 2$ is composite if it is even?

    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.

25. Identify whether $49$ is prime or composite.

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

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