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

Question

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?

A. 8
B. 12
C. 15
D. 16
E. 30

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GMATinsight · Accepted Answer

GCD of (x, y) = 35

i.e. x = 35a
y = 35b

where a and b share NO common factors other than 1

Also, 35a + 35b = 1085

i.e. a +b = 31

(a, b) can be
(1, 30)
(2, 29)
(3, 28)
---
---
and so on
(30, 1)

i.e. 30 pairs cause 31 (Prime) is not divisible by any number other than 1 and 31 so when 31 is broken in two parts then a and b can NOT share any common factor other than 1

Answer: Option E

nick1816 · Answer

x= 35m
y= 35n

where m and n are coprimes

m+n= 1085/35 = 31

Since 31 is a prime, m can be all positive integers less than 31 or m can take 30 values.

E

SushantSaini · Answer

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.?

Kinshook · Answer

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

Fdambro294 · Answer

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…

Posted from my mobile device

bumpbot · Answer

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

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

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