If both x and y are positive integers that are divisible by

Question

If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

(1) x = 4y
(2) x - 3y = 10

Bunuel · Accepted Answer

If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

Given:  x=10m  and  y=10n , for some positive integers  m  and  n .

(1) x = 4y. If  x=40  and  y=10 , then the answer is YES but  x=80  and  y=20 , then the answer is NO. Not sufficient.

(2) x - 3y = 10 -->  10m-3*(10n)=10  -->  m-3n=1  -->  m=3n+1 .  m  and  3n  are consecutive integers. Any two consecutive positive integers are co-prime, which means that they do not share any common factor but 1. For example: 3 and 4, 5 and 6, 100 and 101, are consecutive integers and they do not share any common factor bu 1. So,  m  and  3n  do not share any common factors but 1, which means that  m  and  n  also do not share any common factors but 1, therefore the greatest common factor of  x=10m  and  y=10n  is 10. Sufficient.

Answer: B.

Hope it's clear.

jagdeepsingh1983 · Answer

If both x and y are positive integers that are divisible by 10, is 10 the greatest common factor of x and y?

(1) x = 4y
(2) x - 3y = 10

1. x=4y tells us x is multiple of 4 not we not sure about Y, but both divisible by 10-Not sufficient 
2 x-3y=10, means x is 10 number greater than y but it can be if x 130 y can be 120 but not sure Not Sufficient 
 1010-1000, 210-200 not sufficient

Now combining both we get Y=10 & x=40 which means 10 is HCF

Remember there is difference between LCM & HCF

Berbatov · Answer

I think you forgot the "3 times" in (2)

Capricorn369 · Answer

B is my answer.

Reason -

1. x = 4y 
if x = 40, y =10 -> GCF = 10
if x = 80, y =20 -> GCF = 20 ----- not sufficient.

(2) x - 3y = 10
if y = 10, y= 40 -> GCF = 10
if y = 20, y= 70 -> GCF = 10
if y = 30, y= 100 -> GCF = 10 ------- Sufficient.

Hope it helps.!

jamifahad · Answer

Easiest way to do this is to pick numbers.

Stmt1: x=4y
y=10
x-40
10 is the GCF.
y=100
x=400
100 is the GCF. Yes and No. Not sufficient.

Stmt2: x-3y=10
x=3y+10
y=10
x=40
GCF=10
y=100
x=310
GCF=10
y=50
x=160
GCF=10 Sufficient. 
OA B.

reatsaint · Answer

(1) Putting values, we can have 40,10 or 400,100. GCD can be 10 or 100. NOT SUFFICIENT. 
 (2) x - 3y = 10, again putting values, we can have only 40,10 or 70,20. 
Now let us substitute x = 10a and y = 10b. 
10a – 30b = 10, 
Thus a = (10 +30b) / 10
We can conclude for all values of b, 10 + 30b must be a multiple of 10, hence GCD of x and y is indeed 10. 
SUFFICIENT. 
 Hence ans is B

Daeny · Answer

Is there any way other than substituting values?

EvaJager · Answer

If 10 is the greatest common divisor of  x  and  y  then  x=10a, \, y=10b  for some positive integers  a  and  b , where  a  and  b  are co-prime (meaning their greatest common divisor is 1).

(1) Obviously not sufficient. It just states that  x  is 4 times bigger than  y , which can hold even without the two numbers being divisible by 10.
For example  x=4,\,y=1.

(2) Again, not sufficient.  x=3y+10 ,  x  and  y  not necessarily divisible by 10. For example  x=13,\,y=1.

(1) and (2) together: solving the two equations we obtain unique values for  x  and  y :  x=40,\,y=10. 
Sufficient.

Answer C

sravs27 · Answer

Guess they have mentioned in the question that x & y are divisible by 10. So, x & y cant take values 13 and 1. But still I feel 2nd statement is not sufficient.

EvaJager · Answer

Thanks.
I am such an astronaut...

So, let's try again:

Assume  x=10a  and  y=10b , for some positive integers  a  and  b. 
 x  and  y  have 10 as their greatest common divisor if and only if  a  and  b  are co-prime (their greatest common divisor is 1).

(1)  x=4y  translates into  10a=40 b or  a=4b .
 a  and  b  can be co-prime only if  b=1. 
Not sufficient.

(2)  x-3y=10  becomes  10a-30b = 10  or  a-3b=1 . 
If  a  and  b  have a common divisor  d  (some positive integer), then  a=md  and  b=nd  for some positive integers  m  and  n. 
It follows that  dm-3dn = d(m-3n)=1 , which means that  d  must be a divisor of 1, so  d=1. 
It means that  a  and  b  are co-prime.
Sufficient.

Answer B.

Bunuel · Answer

Bumping for review and further discussion.

suk1234 · Answer

I am Sorry I can't get it, how are     m  and  3n     consecutive integers?

Thanking you in advance!

abhinav11 · Answer

AS Bunuel Proved in the second statement m = 3n + 1 that means m is 1 greater than 3n or m and 3n are consecutive. I hope it is clear now. I am following GMAT quant since 2 months and I am in awe of how Bunuel goes about each question so thoroughly and perfectly. I wish I may achieve half of that. Probably "the greatest" Mathematician I have come across all my life :roll:

Regards, Abhinav

jlgdr · Answer

We can also note on statement 2 that since X and Y are both multiples of 10 and the difference is a multiple of 10 then 10 must be their GCF, if x and y had a higher GCF say like 20,30,40 etc... then the difference of two multiples of 20 will never yield 10 as a difference, same with others Just my 2cents Additional note: If the second statement were x-y = 10 we could also know that the GCF cannot exceed 10 because the GCF is always less than the difference between two numbers and exactly 10 when these two numbers are consecutive integers. Therefore we would know that these two numbers x and y are consecutive integers. But that's for another problem I guess Hope it helps Gimme some freaking Kudos!! J

mvictor · Answer

1. x=40, y = 10 -> GCF = 10 or X=200, y=50. GCF=50 -> A and D out.
2. x-3y=10 or
(x-10)/3 = y
so (x-10)/3 is a number divisible by 10.
x=40 - y=10
x=70 - y=20
x=100 - y=30
so as we can see, GCF always is 10. sufficient.
B is the answer.

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