all positive integer pair (n,r)

juantheron · 2011-10-19 16:30:51

find all positive integer pair of

and in

Where

Bob · 2011-10-19 19:35:11

hi juantheron

I started with a prime decomposition to see what factors we have here.

2012 = 2 x 2 x 503.

So there's not going to be a long string of sequencial factors that lead to 2012.

But every positive whole number is the second number in that line of Pascal's triangle.

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
........................

In other words 2012 = 2012!/(1!.2011!) so n = 2012, r = 1 is one solution.

The other is at the other end of the row. Can you spot it?

Bob

juantheron · 2011-10-20 15:48:13

bob bundy I want more explanation for that.

Bob · 2011-10-20 20:17:54

hi juantherod

I'll have a go.

Pascal's triangle and the combinations formula calculate the same numbers. But, if you haven't heard of Pascal's triangle, don't worry,
I can get to the same place with the combination formula.

Say, n = 20 and r = 7

So you get a long string of numbers one after the other to make the product.

But

503 is a prime so that's it, no more factors.

So you are not going to make a long string combination calculation from that.

But

so every positive whole number can be made if you choose r = 1

Any more answers? Well yes because you can also put n-r = 1, by making r = 2011. Then you get

I'm fairly confident there are no more answers.

Bob

Math Is Fun Forum

#1 2011-10-19 16:30:51

all positive integer pair (n,r)

#2 2011-10-19 19:35:11

Re: all positive integer pair (n,r)

#3 2011-10-20 15:48:13

Re: all positive integer pair (n,r)

#4 2011-10-20 20:17:54

Re: all positive integer pair (n,r)

Board footer