How many ordered pairs of positive integers (x, y) exist such that the

Let’s say the value of a+b comes out to be 32 which is not a prime no. The how the solution would have been different.?

Asked: How many ordered pairs of positive integers (x, y) exist such that the greatest common factor of x and y is 35 and the sum of x and y is 1085?

Let x = 35k
and y = 35m
where k and m are co-prime integers

x + y = 35 (k + m) = 1085
k + m = 31

(k,m) = {(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),,,,, (30,1)} : 30 sets

IMO E

Since we are assuming that

X = 35a

Y = 35b

Where 35 is the GCF of the two numbers X and Y, the values that are inserted for (a) and (b) must NOT share any other prime factors……else the GCF(X ; Y) will not be 35, in violation of the condition in the question stem.

For instance, if somehow we ended up with: a + b = 32

(2 , 30) would be an invalid possibility, since in that case we would put the values in for (a) and (b) and X and Y would have another prime factor of 2 in common ———-> resulting in a GCF of 70.

(4 , 28) inserted into (a) and (b) would be another invalid case ——-> X and Y would instead have a GCF of 35 * 4 since both (a) and (b) would add another TWO primes of 2 to the numbers

I hope the explanation isn’t too muddled…

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