Given that x is a positive integer, determine all possible values of a positive integer constant a such that the product of the digits in the base ten representation of x is equal to x^{2} – ax  22.
*** For an extra challenge, solve this puzzle without using a computer program.
for f(x) = x^2  ax  22
f(a) = 22
f(a+1) = (a+1)^2  a(a+1)  22 = a21
Let p(a) = the product of digits of a
So we need p(a+1) = a21
It is easy to see that a must be 2 digits because a21 gets big too fast. A quick search yields:
a=33, 3321=12=3*4
a=45, 4521=24=4*6
a=56, 5621=35=5*7
a=77, 7721=56=7*8
f(a+2) = 2a18
This time we need p(a+2) = 2a18
a=10, 2*1018=2=1*2
2a18 quickly grows too large
using f(a+3) or higher grows large before a even reaches double digits.
So the solutions for a are {10, 33, 45, 56, 77}

Posted by Jer
on 20111005 11:34:16 