The product of all the prime numbers less than 20 is closest to which of the following powers of 10?

The product of all the prime numbers less than 20 is closest to which of the following powers of 10?

OFFICIAL SOLUTION:

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Their product is 9,699,690 (arrived at as follows:

2 × 3 × 5 × 7 × 11 × 13 × 17 × 19 = 9,699, 690).

This is closest to 10,000,000 = 10⁷

(10 × 10 × 10 × 10 × 10 × 10 × 10 = 10,000,000).

The correct answer is C.

MBA HOUSE SOLUTION:

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, a prime number is a number that cannot be exactly divided by any whole number other than itself and 1. The first few prime numbers are 2, 3, 5, 7, 11, 13, and so forth.

Example Question:

Determine if the number 29 is a prime number.

Solution:

To solve this, you need to check if 29 can be divided evenly by any number other than 1 and itself (29).

Steps:

1. Divisibility Test: Start by testing divisibility by 2. Since 29 is odd, it is not divisible by 2.
2. Continue with Odd Numbers: Continue testing divisibility using the next few smallest prime numbers (3, 5, 7, etc.) up to the square root of 29.
3. Square Root Check: The square root of 29 is approximately 5.4. Therefore, you only need to check divisibility up to the primes less than or equal to 5.
4. 29 is not divisible by 3 (29/3 ≈ 9.67, not an integer).
5. 29 is not divisible by 5 (29/5 = 5.8, not an integer).

Conclusion: Since 29 is not divisible by any of the primes up to 5, and all other factors would have to be greater than the square root and thus pair with factors smaller than the square root (which we have already checked), 29 is confirmed as a prime number.

This example shows the basic process of checking for primality, which is practical for small numbers. For larger numbers, more sophisticated methods and algorithms are usually employed.