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Factors!

by smackmartine » Fri Jun 17, 2011 7:06 pm
If x and y are non zero integers and 450x=120y then which of the following must be an integer?

1)(xy)/60
2)(15x)/(4y)
3)(4x)/(15y)

A) 1 only
B) 2 only
C) 1 and 3
D) 1 and 2
E) 1,2,and 3

Please justify your IMOs ;)
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by Frankenstein » Fri Jun 17, 2011 7:45 pm
Hi,
450x = 120y => 15x = 4y
So, x = (4/15)y. Given that x&y are integers. So, for x to be integer y should be of the form 15k where k is a nonzero integer
for y= 15k, x = 4k
1)(xy)/60 = 15k*4k/60 = k^2(integer)
2)(15x)/(4y) = 1(integer)
3)4x/15y = (4/15)^2(not an integer

Hence, D
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by SUHAILK » Fri Jun 17, 2011 7:47 pm
Answer is D

we know 450x = 120y ( x & y are non zero integers)

on simplifying above relation we get 15x = 4y

or x = 4y/15 and y = 15x/4

since x and y both are integers 1) y should be a multiple of 15 & 2)x should be a multiple of 4

hence xy/60 is integer & 15x/4y = 1 (is a integer)

I hope this was helpful
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by smackmartine » Fri Jun 17, 2011 7:50 pm
1) y should be a multiple of 15 & 2)x should be a multiple of 4

hence xy/60 is integer


OA is D.Thanks. I missed this part.
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by amit2k9 » Fri Jun 17, 2011 8:28 pm
450x = 120y means 15x = 4y
taking LCM of 15 and 4 we have 60.

thus 15*4 = 4*15 xy/60 = true

also, y/x= 15/4. true

thus D it is.
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