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# If x and y are integers each with absolute value less than 5 and 7x -

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Math Expert
Joined: 02 Sep 2009
Posts: 56300
If x and y are integers each with absolute value less than 5 and 7x -  [#permalink]

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29 Dec 2016, 04:59
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55% (hard)

Question Stats:

63% (02:20) correct 37% (02:08) wrong based on 112 sessions

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If x and y are integers each with absolute value less than 5 and 7x - 6y = 3, how many values of x are there?

A. 1
B. 2
C. 3
D. 4
E. 5

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Re: If x and y are integers each with absolute value less than 5 and 7x -  [#permalink]

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29 Dec 2016, 08:28
7x-6y=3
7x=6y+3
Only x= -3 and x=3 satisfy the equation
So
Ands is B ie 2

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Joined: 02 Aug 2009
Posts: 7764
Re: If x and y are integers each with absolute value less than 5 and 7x -  [#permalink]

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29 Dec 2016, 08:57
Bunuel wrote:
If x and y are integers each with absolute value less than 5 and 7x - 6y = 3, how many values of x are there?

A. 1
B. 2
C. 3
D. 4
E. 5

Hi,

A way to do such Qs..
7x-6y=3...
7x=6y+3=3(2y+1)....
We can see from this that x HAS to be a multiple of 3..
Possible values are 3&-3..

But remember to check:-
if these values have y as an integer..
2y+1 can take any ODD values
And for X=3, y=3..
X=-3, y is -4..
So both are OK...
B
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If x and y are integers each with absolute value less than 5 and 7x -  [#permalink]

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29 Dec 2016, 09:23
chetan2u wrote:
Bunuel wrote:
If x and y are integers each with absolute value less than 5 and 7x - 6y = 3, how many values of x are there?

A. 1
B. 2
C. 3
D. 4
E. 5

Hi,

A way to do such Qs..
7x-6y=3...
7x=6y+3=3(2y+1)....
We can see from this that x HAS to be a multiple of 3..
Possible values are 3&-3..

But remember to check:-
if these values have y as an integer..
2y+1 can take any ODD values
And for X=3, y=3..
X=-3, y is -4..
So both are OK...
B

Hi chetan2u, nice explanation, thank you.

Can you explain this part again if you don't mind?
Quote:
But remember to check:-
if these values have y as an integer..
2y+1 can take any ODD values
And for X=3, y=3..
X=-3, y is -4..
So both are OK...

and this
Quote:
We can see from this that x HAS to be a multiple of 3..
Possible values are 3&-3..

I've a sort of weakness with this type of questions :/ Any theory you can suggest in this regard?
_________________
Yours,
Siva Rama Krishna Meka
Math Expert
Joined: 02 Aug 2009
Posts: 7764
Re: If x and y are integers each with absolute value less than 5 and 7x -  [#permalink]

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29 Dec 2016, 18:54
Sirakri wrote:
chetan2u wrote:
Bunuel wrote:
If x and y are integers each with absolute value less than 5 and 7x - 6y = 3, how many values of x are there?

A. 1
B. 2
C. 3
D. 4
E. 5

Hi,

A way to do such Qs..
7x-6y=3...
7x=6y+3=3(2y+1)....
We can see from this that x HAS to be a multiple of 3..
Possible values are 3&-3..

But remember to check:-
if these values have y as an integer..
2y+1 can take any ODD values
And for X=3, y=3..
X=-3, y is -4..
So both are OK...
B

Hi chetan2u, nice explanation, thank you.

Can you explain this part again if you don't mind?
Quote:
But remember to check:-
if these values have y as an integer..
2y+1 can take any ODD values
And for X=3, y=3..
X=-3, y is -4..
So both are OK...

and this
Quote:
We can see from this that x HAS to be a multiple of 3..
Possible values are 3&-3..

I've a sort of weakness with this type of questions :/ Any theory you can suggest in this regard?

Hi
If we have co-prime numbers and x and y are integers then those variables x and y will be multiple of those PRIME numbers..
Say 7x = 8y...Here x will be multiple of 8 and y will be multiple of 7

Similarly here when you say 7x=3(2y+1)....
In 7x, there has to be a multiple of 3 as RHS is a multiple of 3... So x is a multiple of 3
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Re: If x and y are integers each with absolute value less than 5 and 7x -  [#permalink]

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29 Dec 2016, 20:33
I solved it in this way.

We know that x and y are integers such that -5<= x,y<=5
For the equation to hold true 7x - 6y = 3.
1. x and y can not be equal.
2. 6y will always be even so to get the overall difference as odd number 7x should be odd number. Hence x should be odd number. So |x| = 1,3,5

By plug n play we can figure out that 1 and 5 are not possible. Only 3 makes sense. So 2 values 3,-3.

Hope it helps..
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Re: If x and y are integers each with absolute value less than 5 and 7x -  [#permalink]

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05 Mar 2018, 19:13
I used a more brute-force approach, but it's plausible for such question:

7*x - 6*y = 3 => x = (6*y + 3)/7

since both x and y are integers, 6*y + 3 must be a multiple of 7
trying the first 4 multiple of 7 on both positive and negative spectrum, we found that only x = 3 and x = -3 satisfy the condition

Re: If x and y are integers each with absolute value less than 5 and 7x -   [#permalink] 05 Mar 2018, 19:13
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