Le Monde problem: “The prediction”

This week’s Le Monde mathematical problem is:

The scene is set in 2010. Young N, an integer looking for his soul-mate, consults a fortune-teller, who says: “Yes, you shall encounter the beloved number [call it X], a number strictly greater than you, but later! That year [call it Y], you will be the only [my emphasis]  two positive integers whose sum is equal to the year, and whose product is a multiple of the year.” When (at the earliest) will N meet his soul -mate?

This is the kind of question which could be asked at the FFJM semi-finals in a couple of weeks (which my schedule doesn’t allow me to attend, unfortunately). The word only is key. (Christian solves the problem without that word.)

We have, for some integer k,


and we want to minimize Y.

Write Y=ab^2, with a and b integers and b as large as possible. Then it is clear that both X and N are multiples of ab. Write X=xab and N=nab. Suppose that we have a solution to the problem, with (x,n)\neq (1,2); then the couple (x+1,n-1) would give another solution. Hence X=ab, N=2ab and ab^2=Y=X+N=3ab, so b=3.

The problem is now to find the smallest Y=9a>2010 such that no perfect square divides a (otherwise b would not be as large as possible). 2016=9\times 4^2 \times 14 and 2025=9\times 15^2 don’t work, so we move to 2034=9\times 2\times 113, which does work.

The solution is then that in year Y=2034, young N=678 will meet X=1356 and they will live happily ever after.



One Response to “Le Monde problem: “The prediction””

  1. robinryder Says:

    Actually, X and N need only be multiples of the least common multiple of a and b, rather than of ab. Luckily, that doesn’t matter in this case.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: