Prime and Composite Numbers

Show your work for each problem.

1.
Which of these numbers is prime?
1. a) $21$
2. b) $11$
3. c) $9$
4. d) $15$
Answer: $11$
$11$ is prime because its only factors are $1$ and $11$ . The others are composite: $9 = 3 \times 3$ , $15 = 3 \times 5$ , $21 = 3 \times 7$ .
2.
Which statement about the number $1$ is correct?
1. a) $1$ is neither prime nor composite
2. b) $1$ is a composite number
3. c) $1$ is both prime and composite
4. d) $1$ is a prime number
Answer: $1$ is neither prime nor composite
$1$ has only one factor (itself), so it doesn't meet the definition of prime (exactly two factors) or composite (more than two factors).
3.
What is the smallest prime number?
1. a) $1$
2. b) $3$
3. c) $2$
4. d) $0$
Answer: $2$
$2$ is the smallest prime number. It has exactly two factors: $1$ and $2$ . It's also the only even prime number!
4.
Which of these numbers is composite?
1. a) $12$
2. b) $7$
3. c) $13$
4. d) $17$
Answer: $12$
$12$ is composite because it has factors $1, 2, 3, 4, 6, 12$ . The numbers $7$ , $13$ , and $17$ are all prime.
5.
How many factors does $19$ have? Use your answer to decide whether $19$ is prime or composite.
Answer: 2
$19$ is not divisible by $2$ (it's odd) or by $3$ ( $1+9=10$ , not divisible by $3$ ). Since $5 \times 5 = 25 > 19$ , we don't need to check further. So $19$ has exactly $2$ factors, $1$ and $19$ : it is prime.
6.
How many prime numbers are there between $1$ and $10$ ?
Answer: 4
The primes between $1$ and $10$ are: $2, 3, 5, 7$ . That's $4$ prime numbers.
7.
What is the next prime number after $23$ ?
Answer: 29
$24 = 2 \times 12$ , $25 = 5 \times 5$ , $26 = 2 \times 13$ , $27 = 3 \times 9$ , $28 = 4 \times 7$ . But $29$ has only factors $1$ and $29$ , so it's prime!
8.
How many factors does the number $18$ have?
Answer: 6
The factors of $18$ are: $1, 2, 3, 6, 9, 18$ . That's $6$ factors, so $18$ is composite.
9.
Determine if $37$ is prime or composite.
Answer: prime
1. What is the largest whole number whose square is at most $37$ ? (This is $\sqrt{37}$ rounded down.) 6
2. Is $37$ divisible by $2$ ? no
3. Is $37$ divisible by $3$ ? no
4. Is $37$ divisible by $5$ ? no
5. Is $37$ prime or composite? prime
10.
Find all the factors of $36$ and determine if it's prime or composite.
Answer: composite
1. Is $36$ divisible by $1$ ? yes
2. Is $36$ divisible by $2$ ? If yes, what's $36 \div 2$ ? 18
3. Is $36$ divisible by $3$ ? If yes, what's $36 \div 3$ ? 12
4. Is $36$ divisible by $4$ ? If yes, what's $36 \div 4$ ? 9
5. Is $36$ divisible by $6$ ? If yes, what's $36 \div 6$ ? 6
6. The factors of $36$ are $1, 2, 3, 4, 6, 9, 12, 18, 36$ . Is $36$ prime or composite? composite
11.
List all prime numbers between $20$ and $30$ .
Answer: 23, 29
1. Is $21$ prime? no
2. Is $22$ prime? no
3. Is $23$ prime? yes
4. Is $25$ prime? no
5. Is $27$ prime? no
6. Is $29$ prime? yes
12.
Which of the following statements is TRUE?
1. a)All prime numbers are odd
2. b)All even numbers greater than $2$ are composite
3. c)All odd numbers are prime
4. d)The number $51$ is prime
Answer: All even numbers greater than $2$ are composite
Every even number > $2$ is divisible by $2$ , so it has at least $3$ factors ( $1$ , $2$ , itself). $2$ is prime (and even). $9$ and $15$ are odd but composite. $51 = 3 \times 17$ .
13.
What is the sum of all prime numbers between $10$ and $20$ ?
Answer: 60
The primes between $10$ and $20$ are $11, 13, 17, 19$ . Sum: $11 + 13 + 17 + 19 = 60$ .
14.
Is $91$ prime or composite? Show your work.
Answer: composite
1. What is the largest whole number whose square is at most $91$ ? (This is $\sqrt{91}$ rounded down.) 9
2. Is $91$ divisible by $2$ ? no
3. Is $91$ divisible by $3$ ? no
4. Is $91$ divisible by $5$ ? no
5. Is $91$ divisible by $7$ ? What is $91 \div 7$ ? 13
6. Is $91$ prime or composite? composite
15.
Express $60$ as a product of prime factors.
Answer: $60 = 2^2 × 3 × 5$
1. Is $60$ divisible by $2$ ? What is $60 \div 2$ ? 30
2. Is $30$ divisible by $2$ ? What is $30 \div 2$ ? 15
3. Is $15$ divisible by $2$ ? no
4. Is $15$ divisible by $3$ ? What is $15 \div 3$ ? 5
5. Is $5$ prime? yes
6. $60 = 2^{\square} \times 3 \times 5$ . What exponent goes in the box? 2