SOLUTION: A pronic number is the product of two consecutive positive integers. 110 is a pronic number as 110 = 11 x 10. The sum of two pronic numbers is 240. What is the difference betwee

Algebra ->  Sequences-and-series -> SOLUTION: A pronic number is the product of two consecutive positive integers. 110 is a pronic number as 110 = 11 x 10. The sum of two pronic numbers is 240. What is the difference betwee      Log On


   

Question 1006021: A pronic number is the product of two consecutive positive integers.
110 is a pronic number as 110 = 11 x 10.
The sum of two pronic numbers is 240.
What is the difference between these two numbers?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
There may be simpler, more efficient ways to solve this, and I would love to hear about them, but here is what I could come up with.

I would look for two positive integers, A and B , such that
A and B are pronic numbers,
A%3EB , and A%2BB=240 .
Since 240=A%2BB%3E2B%29--->120%3EB%29 and since 11%2A12=132%3E120 , B%3C=10%2A11=110 .
So, the smallest of the two pronic numbers we are looking for is one of the 10 smallest pronic numbers.

BRUTE FORCE SOLUTION:
A brute force solution (trying the first ten pronic numbers) looks feasible.
We could try each of the pronic numbers numbers from B=1%2A2=2 to B=10%2A11=110
and see if 240-B%3C240 is a pronic numbers number .


With a list of pronic numbers that would be easy.

With a calculator, I would add the digits 25 to the right of each 240-B number,
and take the square root to see if it is an integer.
That is a known property of pronic numbers.
I did not know that, but I took the square roots of the 240-B%2B0.25 ,
until I found that sqrt%28210.25%29=14.5 ,
which is the same as sqrt%2821025%29=145
I used the fact that if 240-B is pronic,
there is a positive integer n such and n%28n%2B1%29=240-B
n%28n%2B1%29=240-B
%28n%2B0.5-0.5%29%28n%2B0.5%2B0.5%29=240-B
%28n%2B0.5%29%5E2-0.5%5E2=240-B
%28n%2B0.5%29%5E2-0.25=240-B
%28n%2B0.5%29%5E2=240-B%2B0.25 .

A MORE ELEGANT SOLUTION:
Without a calculator,
I figured I was looking for the n%5Eth and p%5Eth pronic numbers,
n%2A%28n%2B1%29=%28n%2B0.5%29%5E2-0.25 and p%2A%28p%2B1%29=%28p%2B0.5%29%5E2-0.25 ,
with n and p positive integers, such that
%28n%2B0.5%29%5E2-0.25%2B%28p%2B0.5%29%5E2-0.25=240
%28n%2B0.5%29%5E2%2B%28p%2B0.5%29%5E2=240%2B0.5
4%28n%2B0.5%29%5E2%2B4%28p%2B0.5%29%5E2=4%2A240%2B4%2A0.5
%282n%2B1%29%5E2%2B%282p%2B1%29%5E2=962 .
So, I need two odd numbers whose squares add to 962 .
Since odd numbers' squares end in 1, 5, or 9,
to add up to a sum ending in 2,
both squares should end in 1.
If the p%5Eth pronic number is the smallest of the two we are looking for,
and we had initially figured out that p%3C=10 ,
the only possibilities are
2p%2B1=11--->p=5 and 2p%2B1=21--->p=10 .
2p%2B1=11--->%282p%2B1%29%5E2=121--->%282n%2B1%29%5E2=962-121=841--->2n%2B1=sqrt%28841%29=29
That gives us the solution:
and
.
The other choice,
2p%2B1=21--->%282p%2B1%29%5E2=441--->%282n%2B1%29%5E2=962-441=521 ,
does not work, because 521 is not a perfect square.