Question

Given three different prime numbers
`p_1 \ \ ; \ \ p_2` and
`p_3` satisfy equation
`p_1+p_2+p_3=202` . Find the maximum value of


`\pmb {p_1 * p_2* p_3=?}`

Answer 1

• 19982

Explanation:

As we know all primes but 2 are odd numbers.

It means sum of any 3 primes not including 2 is odd.

Since we have sum of 202, one of our primes is 2.

Sum of the other two primes is 200.

In order to have maximum value of p*(200 - p) these two numbers must be closer to each other. So we are looking for two primes around 100.

We can test all primes less than 200:

• 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, and 199

The 3 primes with max product are:

• 2, 97, 103

The value is:

• 2*97*103 = 19982