Prime and Composite Numbers Practice Problems Online

To determine whether the number $n = 29$ is prime or composite, we will check if $n$ has any divisors other than 1 and itself.

The definition of a prime number is one that has exactly two distinct positive divisors: 1 and itself. Conversely, a composite number has more than two distinct positive divisors.

First, observe that the square root of 29 is approximately 5.385. This tells us that we only need to check divisibility by all prime numbers less than or equal to 5. These primes are 2, 3, and 5.

• Check divisibility by 2: 29 is odd, so it is not divisible by 2.
• Check divisibility by 3: The sum of the digits of 29 is $2 + 9 = 11$, which is not divisible by 3, so 29 is not divisible by 3.
• Check divisibility by 5: 29 does not end in 0 or 5, so it is not divisible by 5.

Since 29 is not divisible by any of these primes, it has no divisors other than 1 and 29 itself. Therefore, 29 is a prime number.

Hence, the solution to the problem is that the number $n = 29$ is Prime.

To determine if the number 19 is prime, follow these steps:

• Step 1: Check if the number is greater than 1. Since $19 > 1$, proceed to the next step.
• Step 2: Identify potential divisors for 19 by considering integers from 2 up to $\lfloor \sqrt{19} \rfloor$.

The square root of 19 is approximately 4.36, and thus we test divisibility by integers 2, 3, and 4.

• 19 divided by 2: The quotient is not an integer (it gives 9.5).
• 19 divided by 3: The quotient is not an integer (it gives 6.333...).
• 19 divided by 4: The quotient is not an integer (it gives 4.75).

None of these divisions result in an integer, meaning 19 has no divisors other than 1 and 19 itself.

Therefore, the number 19 is prime.

To determine whether $n = 23$ is a prime number, we will test its divisibility:

• Step 1: Calculate $\sqrt{23}$. The approximate value is 4.795, and thus we consider prime numbers up to the integer part, which is 4.
• Step 2: Check if 23 is divisible by any prime numbers less than or equal to 4. These primes are 2 and 3.

Step 3: Test divisibility:
- 23 is not divisible by 2, as it is odd.
- 23 is not divisible by 3, since $23 \div 3 \approx 7.67$, which is not an integer.

Since 23 is not divisible by any prime number less than or equal to its square root, it only has divisors of 1 and 23. Hence, 23 is a prime number.

Therefore, the solution to the problem is that $n = 23$ is prime.

A number is classified as prime if it has exactly two distinct positive divisors: 1 and itself. Conversely, a number is composite if it has more than two divisors.

Given the number $n = 10$, we need to determine whether it is prime or composite.

Let's test the divisibility of 10 by numbers other than 1 and 10:

• Check divisibility by 2: Since 10 is an even number, it is divisible by 2. Specifically, $10 \div 2 = 5$ with no remainder.
• Check divisibility by 3: $10 \div 3 \approx 3.333$, which is not an integer, hence not divisible.
• Check divisibility by 5: $10 \div 5 = 2$ with no remainder, showing divisibility by 5.

Since 10 is divisible by numbers other than 1 and itself (specifically 2 and 5), it is not prime. Therefore, the number 10 is composite.

In conclusion, the number 10 is a composite number.

To solve this problem, we'll determine if 42 is a prime or composite number by checking its divisibility by numbers other than 1 and itself.

A number is prime if it has exactly two distinct positive divisors: 1 and itself. It is composite if it has more than two distinct divisors.

Let's find the divisors of 42:

• $42 \div 1 = 42$
• $42 \div 2 = 21$ (evenly divisible, so 2 is a divisor)
• $42 \div 3 = 14$ (evenly divisible, so 3 is also a divisor)
• $42 \div 6 = 7$ (evenly divisible, so 6 is another divisor)
• $42 \div 7 = 6$ (evenly divisible, so 7 is a divisor)
• $42 \div 14 = 3$ (evenly divisible, so 14 is a divisor)
• $42 \div 21 = 2$ (evenly divisible, so 21 is a divisor)
• $42 \div 42 = 1$

From the above list, we can see that 42 has divisors other than 1 and itself, namely 2, 3, 6, 7, 14, and 21. This means that 42 is not a prime number.

Therefore, the number 42 is a composite number.