Is 1681 a prime number?

Steps to Solve

1. Identify the Definition of a Prime Number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

2. List Potential Factors

To check if 1681 is prime, we will look for divisors. We only need to check for factors up to the square root of 1681.

First, calculate the square root:

$$ \sqrt{1681} \approx 41 $$

Thus, we will check for factors from 2 to 41.

3. Check Divisibility

We will check if 1681 is divisible by each integer from 2 to 41. If we find any divisor, then 1681 is not a prime number.

• Check divisibility by 2: $1681$ is odd, so it's not divisible by $2$.
• Check divisibility by 3: Sum of digits $1 + 6 + 8 + 1 = 16$ is not divisible by $3$.
• Check divisibility by 4: Last two digits, $81$, are not divisible by $4$.
• Continue this process up to $41$.

4. Discover the Factorization

When checking divisibility, we find that:

$$ 1681 = 41 \times 41 $$

This means that 41 multiplied by itself equals 1681.

5. Conclusion

Since we found a divisor other than 1 and 1681, we conclude that 1681 is not a prime number.