My Favorite Number
My favorite number is a positive integer. It can be expressed as where are all positive integers, with 2 of them being primes and the other being a composite number.
It is interesting to note that the sum of the divisors of one of is equal to the sum of the other two integers.
What is my favorite number?
The answer is 37.
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1 solution
Let A be the favoritr number in the system: [ A = B + C * D ] , [A = B * D - C ] , [ A = B * C + |B-C| ]
For B>C, take equation (1)+(2)-(3), we'll obtain:
A= CD+BD-BC as equation (4)
Now compare (1) with (4), we'll get:
CD+B = CD+BD-BC
B=B(D-C)
D-C=1
Similarly, compare (4) with (2), we'll get B-D=1. Therefore, B,C,D are consecutive 3 integers.
In case of B<C, we'll obtain: D+C=1, which has no set solutions for natural numbers.
With two of consecutive numbers are prime, those primes must be twin primes. Otherwise, it will include: 2,3,4 which doesn't satisfy the last clue.
For the last clue, such sum of divisors of a prime P = P+1, which can never equal the sum of other two numbers.
That is, P+1 = (P-2)+(P-1), where only P=4, which is not prime.
Thus, such number n must be composite between two primes.
Then sum of n's divisors = (n+1)+(n-1) =2n. Hence, n is a perfect number.
According to digital root , every perfect number greater than 6 has a digital root of 1 (digit sum) and thus not a multiple of 3, but for three consecutive numbers, one of them must be multiple of 3.
Thus, only 5,6,7 works.
By substituting the values, we'll get: A = 6 2 + 1 = 3 7 .