Prime * Prime

Number Theory Level 2

The product of 2 2 distinct prime numbers is between 10 10 and 50 50 . What is the maximum possible value of their sum?


The answer is 25.

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Chung Kevin by Chung Kevin , 45, Australia

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6 solutions

The left table shows the products of two primes and those between 10 and 50 are shaded green. The right table shows the sum of the pair of eligible primes. And the maximum of the sum is found to be 25 \boxed{25} .

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Thanks for the detailed analysis.

With a bit of work, we can show that for large enough numbers, the answer will always be of the form 2 + p 2 + p . This is because for a large enough n n , there is a prime between n n and 3 2 n \frac{3}{2} n .

Chung Kevin - 6 years, 3 months ago
Akash Gupta
Feb 2, 2015

The least prime number is 2. So to get maximum sum but minimum product the number is to be multiplied by 2.. Furthermore, 50/2=25. So, the biggest prime number less than 25 is the second no. which is 23. So, their sum is 25.

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How do you know that we cannot do better?

Is it always true that the maximum sum must involve 2 as a prime?

Chung Kevin - 6 years, 4 months ago

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The maximum possible sum of two numbers, in any given range, always ends up to be the sum of the smallest and the largest number. Since 2 is the smallest prime, the sum involves 2....

Aishwarya Nagendra - 6 years, 3 months ago

Only 2 is even prime. All others must be odd . For our condition, the number can not have more than two factors. Starting with 50 and decreasing, 50 can have to three factors. 49 the primes are notare not distinct. 48 can have more than two factors. 47, no factors. 46 = 2 * 23, both prime and we have taken from the larger values. So 2+23=25. Check:-45,44 can have more than two factors. 43 itself a prime. 42, more factors. 43 itself a prime. 42, more than two. 41, itself a prime. 40, more than two. 39=3 * 13 , sum 16 less than 25... so otheres sum would be less than 25. 25 maximum checked.

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Apart from 50 itself, 25 is the biggest whole number that can be multiplied to give 50. To satisfy the given conditions, we pick the largest prime before 25, which is 23. The biggest other prime that will keep the product below 50 is 2. So we have 2 and 23, whose sum is 25.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

The maximum possible sum of two numbers, in any given range, always ends up to be the sum of the smallest and the largest number. Here, the smallest prime number is two and the product of the two prime numbers should not exceed 50. So, the other prime number is the one that comes immediately before 25(since 25*2 is 50) and voila! it is 23. Hence the sum becomes 23+2=25. Do upvote if you like this solution.....

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Not necessarily. Note that we are restricting our attention to just the prime numbers, and not all reals / integers. So, how do you know that there aren't 2 better primes to use?

Chung Kevin - 6 years, 2 months ago
Raven Herd
Jan 31, 2015

product is 46 and sum is 23 + 2=25.

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How do you know that we cannot do better?

Chung Kevin - 6 years, 4 months ago

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