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
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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) (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, and , such that
and are pronic numbers,
, and .
Since ---> and since , .
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 to
and see if is a pronic numbers number .
With a list of pronic numbers that would be easy.
With a calculator, I would add the digits to the right of each 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 ,
until I found that ,
which is the same as
I used the fact that if is pronic,
there is a positive integer such and
.
A MORE ELEGANT SOLUTION:
Without a calculator,
I figured I was looking for the and pronic numbers,
and ,
with and positive integers, such that
.
So, I need two odd numbers whose squares add to .
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 pronic number is the smallest of the two we are looking for,
and we had initially figured out that ,
the only possibilities are
---> and ---> .
--->--->--->
That gives us the solution:
and
.
The other choice,
--->---> ,
does not work, because is not a perfect square.
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