Question

The sum of two prime numbers and an odd positive integer is 128. What is the largest of the possible primes?

Answer 1

Prime numbers are numbers that can only be divided by itself and 1. The largest possible prime number that fits the scenario is 113

Let the prime numbers be p1 and p2, where p1 > p2; and the odd numbers be x1 and x2

So, we have:

$p_1 + p_2 + x_1 + x_2 = 128$

The largest prime number less than 128 is 127.

If $p_1 = 127$, then

$p_1 + p_2 + x_1 + x_2 = 128$ becomes

$127 + p_2 + x_1 + x_2 = 128$

$p_2 + x_1 + x_2 = 128-127$

$p_2 + x_1 + x_2 = 1$

This is not possible, because three positive integers cannot add up to 1

The next largest prime number is 113

If $p_1= 113$, then

$p_1 + p_2 + x_1 + x_2 = 128$ becomes

$113 + p_2 + x_1 + x_2 = 128$

Collect like terms

$p_2 + x_1 + x_2 = 128-113$

$p_2 + x_1 + x_2 = 15$

Let $p_2 = 3$

$p_2 + x_1 + x_2 = 15$ becomes

$3 + x_1 + x_2 = 15$

Collect like terms

$x_1 + x_2 = 15-3$

$x_1 + x_2 = 12$

$x_1$ and $x_2$ are odd numbers.

So, we have:

$x_1 = 5\\x_2 =7$

This is true because

$x_1 + x_2 = 12$

$5 + 7 = 12$

Hence, the largest of the possible primes is 113

Read more about prime numbers at:

brainly.com/question/4184435