Andrew arranged the number 1, 2, 3, ...11, 12 into six pairs so that the sum ofany two numbers in a pair is prime and no two of these primes are equal. find the primes and the pairs
I had to solve this by "trial and error" and finally came up with this. There many be other solutions.
1 + 4 = 5
2 + 5 = 7
3 + 8 = 11
6 + 7 = 13
9 + 10 = 19
11 + 12 = 23
Other prime-number-sum combinations possible would have been 3 and 17, but by forming them I had to use numbers twice or get duplicate sums.
(5,8);(3,2);(1,6);(9,10);(4,7);(11,12).
5+8=13
3+2=5
1+6=7
9+10=19
4+7=11
11+12=23
Hope that helps!!!!! :)
Well, Andrew certainly seems to have taken quite a mathematical challenge!
To find the pairs, let's start by listing all the primes between 1 and 12: 2, 3, 5, 7, 11.
Now, let's see if we can make pairs that satisfy the conditions:
Pair 1: 2 and 3. Their sum is 5, which is prime.
Pair 2: 5 and 7. Their sum is 12, which is not prime.
Pair 3: 11 and 12. Their sum is 23, which is prime.
Pair 4: 2 and 5. Their sum is 7, which is prime.
Pair 5: 3 and 7. Their sum is 10, which is not prime.
Pair 6: 11 and 6. Their sum is 17, which is prime.
So, the pairs that satisfy the conditions are:
Pair 1: 2 and 3
Pair 2: 11 and 12
Pair 3: 2 and 5
Pair 4: 11 and 6
Remember, this is just one possible solution. There may be other combinations that also work.
To find the primes and pairs that satisfy the given conditions, we need to identify pairs of numbers whose sum is prime, and ensure that no two pairs share the same prime sum.
Let's start by listing all the prime numbers up to the maximum possible sum of any pair in the range from 1 to 12. The largest possible sum would be 12 + 11 = 23, so we only need to consider prime numbers up to 23.
Prime numbers up to 23: 2, 3, 5, 7, 11, 13, 17, 19, 23.
Now, let's create all the possible pairs using these numbers:
Pair 1: (2, 3) - Sum: 2 + 3 = 5 (prime)
Pair 2: (2, 5) - Sum: 2 + 5 = 7 (prime)
Pair 3: (3, 4) - Sum: 3 + 4 = 7 (prime)
Pair 4: (5, 8) - Sum: 5 + 8 = 13 (not prime)
Pair 5: (7, 10) - Sum: 7 + 10 = 17 (prime)
Pair 6: (11, 12) - Sum: 11 + 12 = 23 (prime)
From the above pairs, we can see that pairs 1, 2, 3, 5, and 6 have prime sums and no two pairs share the same prime.
So, the primes and pairs that satisfy the given conditions are:
Primes: 5, 7, 17, 23
Pairs: (2, 3), (2, 5), (3, 4), (7, 10), (11, 12)
1+2=3
3+4=7
5+8=11
6+7=13
9+10=19
11+12+23