How many sets of 3 primes sum to 20?
1 isn't prime- no offense Damon, so only 2 sets...
To find out how many sets of 3 primes sum to 20, we need to consider all possible combinations of prime numbers whose sum is 20. Let's go through the numbers and check for prime combinations:
1. Start with the smallest prime number, 2.
- Since 2 is the only even prime number, we cannot have a sum of 20 using three 2s.
2. Next, consider the next smallest prime number, 3.
- If we use three 3s (3 + 3 + 3), the sum is 9, which is less than 20.
- If we use two 3s and one 5 (3 + 3 + 5), the sum is 11, which is less than 20.
- If we use one 3 and two 7s (3 + 7 + 7), the sum is 17, which is less than 20.
- If we use one 3, one 5, and one 7 (3 + 5 + 7), the sum is 15, which is less than 20.
- If we use one 3, one 7, and one 11 (3 + 7 + 11), the sum is 21, which is greater than 20.
Since none of these combinations yield a sum of exactly 20, there are no sets of 3 prime numbers that sum to 20.
To find out how many sets of 3 primes sum to 20, we can start by listing all the possible combinations of 3 primes that sum to 20 and then counting them.
First, let's identify all the prime numbers less than or equal to 20:
2, 3, 5, 7, 11, 13, 17, 19
Next, we can generate all possible combinations of these primes, taking three primes at a time. There are a few different methods to generate these combinations, but let's use a simpler approach called brute force with nested loops:
```
count = 0
for i in range(len(primes)):
for j in range(i+1, len(primes)):
for k in range(j+1, len(primes)):
if primes[i] + primes[j] + primes[k] == 20:
count += 1
```
In the code above, we initialize a counter variable `count` to keep track of the number of sets that sum to 20. The three nested loops iterate through all possible combinations of three primes from the given list. If the sum of the three primes is equal to 20, we increment the `count`.
Let's run this code to get the answer:
```
primes = [2, 3, 5, 7, 11, 13, 17, 19]
count = 0
for i in range(len(primes)):
for j in range(i+1, len(primes)):
for k in range(j+1, len(primes)):
if primes[i] + primes[j] + primes[k] == 20:
count += 1
print(count)
```
When you run this code, it will output the answer: the number of sets of 3 primes that sum to 20.
Note: This specific problem can be solved relatively quickly since the number of primes less than or equal to 20 is small. However, for larger numbers, the brute-force approach becomes computationally expensive, and more efficient algorithms should be employed.
17, 1 and 2
13, 2 and 5
11, 2 and 7
are there others ?