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If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?

1) x = 12u, where u is an integer.

2) y = 12z, where z is an integer.

well I guess the first questions was quite easy. How about this one? do you still use numbers to solve?

OK. Algebraic approach:

Given: x=8y+12.

(1) x=12u --> 12u=8y+12 --> 3(u-1)=2y --> the only thing we know from this is that 3 is a factor of y. Is it GCD of x and y? Not clear: if x=36, then y=3 and GCD(x,y)=3 but if x=60, then y=6 and GCD(x,y)=6 --> two different answers. Not sufficient.

(2) y=12z --> x=8*12z+12 --> x=12(8z+1) --> so 12 is a factor both x and y.

Is it GCD of x and y? Why can not it be more than 12, for example 13, 16, 24, ... We see that factors of x are 12 and 8z+1: so if 8z+1 has some factor >1 common with z then GCD of x and y will be more than 12 (for example if z and 8z+1 are multiples of 5 then x would be multiple of 12*5=60 and y also would be multiple of 12*5=60, so GCD of x and y would be more than 12). But z and 8z+1 CAN NOT share any common factor >1, as 8z+1 is a multiple of z plus 1, so no factor of z will divide 8z+1 evenly, which means that GCD of x and y can not be more than 12. GCD(x,y)=12. Sufficient.

Re: GCD 2 (Tougher) [#permalink]
25 Oct 2010, 07:55

Expert's post

rafi: Same logic as that given by Bunuel and shrouded1 above, just worded differently in case you have come across this before: "Two consecutive integers do not have any common factors other than 1"

So 8z and 8z + 1 will not share any factors other than 1 and all factors of z will be factors of 8z too. Therefore, z and 8z + 1 will not have any common factors other than 1. _________________