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Integral solutions 4x-Ay=B?

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Integral solutions 4x-Ay=B?

by gmatrant » Thu Jul 01, 2010 5:35 pm
The equation 4x-Ay=B has a number of integral solutions. If HCF(A,4)=1
and the no. of solutions (x,y) for 0<x,y<500 is 45, then the number of
possible values of A is
a)3
b)2
c)1
d)0
e)none of these
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by saptrishi » Thu Jul 01, 2010 6:50 pm
This definitely is not a GMAT question.
What is the correct answer?
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by albatross86 » Sat Jul 03, 2010 8:59 am
Interesting, let's have a go for fun:

4x - Ay = B

=> Ay = 4x - B

=> y = 4/A * x - B

There are exactly 45 integer values of x between 0 and 500 that provide integer values of y.

Thus, these are values such that x/A has EXACTLY 45 integer values for some 45 values of x. (Since A and 4 are relatively prime, 4 does not contribute to this aspect)

So, in short, A needs to be an integer that is:
1. Relatively prime to 4.
2. Produces exactly 45 multiples in the range 0 to 500 exclusive.

How many values of A would do this?

For this, we can see that we need a number A, where 45*A < 500 but 46*A > 500

=> A < 11.11 but A > 10.87

The only number that satisfies this is 11.

Hence C - 1 possible value

I'm not sure if this is correct, but anyway was fun :)
~Abhay

Believe those who are seeking the truth. Doubt those who find it. -- Andre Gide
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