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Multiply Digits, Get Quadratic (Posted on 2011-10-05) Difficulty: 3 of 5
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 x2 ax - 22.

*** For an extra challenge, solve this puzzle without using a computer program.

See The Solution Submitted by K Sengupta    
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Solution Different solution Comment 2 of 2 |
for f(x) = x^2 - ax - 22
f(a) = -22
f(a+1) = (a+1)^2 - a(a+1) - 22 = a-21

Let p(a) = the product of digits of a
So we need p(a+1) = a-21
It is easy to see that a must be 2 digits because a-21 gets big too fast.  A quick search yields:
a=33, 33-21=12=3*4
a=45, 45-21=24=4*6
a=56, 56-21=35=5*7
a=77, 77-21=56=7*8

f(a+2) = 2a-18
This time we need p(a+2) = 2a-18
a=10, 2*10-18=2=1*2
2a-18 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 2011-10-05 11:34:16
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