Kassey's favorite number is the integer n . Paul's favorite number is the integer n + 1 .
Given that both n and n + 1 have exactly four divisors and have the same sum of divisors. What is the number of divisors of their product?
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Oh no I didn't thought of this, I found the numbers as 1 4 , 1 5 but not after wasting my 10 minutes.
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Can't the numbers be 27 and 28
Great work @Christopher Boo ..Didn't think of this ..I worked out the numbers to be 14 and 15.
To say the numbers have exactly four divisors means that they are each a product of two distinct prime numbers, say p q and r s . To say the sum of their divisors is the same means 1 + p + q + p q = 1 + r + s + r s , and to say they are consecutive means p q + 1 = r s . Putting the two equations together gives p + q = 1 + r + s , which can happen only when one of the primes is even (hence 2 ). A little trial and error produces some primes that work: p = 2 , q = 7 , r = 3 , and s = 5 . So the two numbers could be 1 4 and 1 5 , both with sum of factors equal to 2 4 . The number of factors of 1 4 ∗ 1 5 = 2 ∗ 3 ∗ 5 ∗ 7 is 2 4 = 1 6 .
Given Christopher's answer, it probably would have been better to have just asked for the least possible sum of the favorite numbers, namely 2 9 . But this does raise an even more interesting question: how many more possibilities are there for the two favorite numbers? I'm finding that we must have that
n = 2 p + p − 2 2 ( p + 1 )
for some prime p , on the condition that, of course, n must be a positive integer. Now both p = 3 and p = 5 yields n = 1 4 , but I don't think that there are any more solutions, i.e., n = 1 4 is the only solution. For p = 7 we don't get an integer, and for primes p > 7 we have that
2 < p − 2 2 ( p + 1 ) < 3 ,
and so we can never obtain an integral value for n other than 1 4 .
@Paul Ryan Longhas I have posted a rephrased version of your question that reflects this last calculation. I hope that's o.k. with you; I have made reference to your question in a comment to my solution of the new question.
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That's exciting. Could you post this version of the problem? Thanks!
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O.k., here it is. :) I'll start playing around with variations of this problem, see if anything interesting comes up.
Lol! We posted same thing at same time. Given you have posted explanation, I can delete my comment.
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Haha. Yeah, I guess once is enough. :) It is an interesting question though to show that there is a unique solution to what the favorite numbers are.
Yes, I can feel that the problem maker wasn't expecting my type of solution. The problem should be stated in a different way such that the answer is non-trivial.
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n and n + 1 are co-prime. That is, they don't share the same divisor except 1 . Hence,
d ( n ) × d ( n + 1 ) = 4 × 4 = 1 6