if x and y are positive integers such that 3x+7y is a multiple of 11 then which of the following will also be dividible by 11:
1. 4x+6y
2. x+y+4
3. 9x+4y
4. 4x -9y
5. 7x + 4y
multiple
This topic has expert replies
Choose values of x and y such that 3x + 7y is a multiple of 11.
x = 5, y = 1 works
Plug each of these into the answers and you'll find that:
1. 4x + 6y = 20 + 6 = 26 WRONG
2. x + y + 4 = 5 + 1 + 4 = 10 WRONG
3. 9x + 4y = 45 + 4 = 49 WRONG
4. 4x - 9y = 20 - 9 = 11
5. 7x + 4y = 35 + 4 = 39 WRONG
Therefore option 4
x = 5, y = 1 works
Plug each of these into the answers and you'll find that:
1. 4x + 6y = 20 + 6 = 26 WRONG
2. x + y + 4 = 5 + 1 + 4 = 10 WRONG
3. 9x + 4y = 45 + 4 = 49 WRONG
4. 4x - 9y = 20 - 9 = 11
5. 7x + 4y = 35 + 4 = 39 WRONG
Therefore option 4
Last edited by moutar on Mon Apr 20, 2009 11:17 am, edited 1 time in total.
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Is there any simpler way other than plugging values?
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There are quite a few ways to see why this works - there's surely a faster way than what I've done below, but it's the end of a long day here!dumb.doofus wrote:Is there any simpler way other than plugging values?
_______
If 3x+7y is a multiple of 11, then 3x + 7y - 22y = 3x - 15y is a multiple of 11 (whenever we add or subtract two multiples of 11, we must get a multiple of 11). So 3x - 15y = 3(x - 5y) is a multiple of 11. Since 3 is not a multiple of 11, x - 5y must be a multiple of 11. We know now that 3x + 7y and x - 5y are multiples of 11; if we add them we must get another multiple of 11:
3x + 7y
x - 5y
4x + 2y
So 4x + 2y is a multiple of 11, and so is 4x + 2y - 11y = 4x - 9y.
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Well, there's not really any 'magic' involved, because if you do any similar sequence of steps to what I did above, you'll arrive at the answer. Of course, it helps a lot to know the correct answer in advance; picking numbers, as moutar did above, seems the best approach in test conditions.maihuna wrote:Ian,
Can you please elaborate further steps on how to pick the magic numbers here in the two steps mentioned by you.
Regards,
maihuna
I only wanted to show *why* the relationship is true, no matter what x and y are. The goal is to rewrite the expression in its simplest form (with the smallest possible numbers). Factoring seems the best way to do that. Looking at 3x + 7y, there is no common factor, but if you add 11x (to get 14x + 7y), or 11y (to get 3x + 18y), for example (there are many other options), you can then factor something out, and get smaller numbers in your expression. So, for example, since 14x + 7y is divisible by 11, and since 14x + 7y = 7(2x + y), then 2x+y must be divisible by 11, since 7 is not. Once you discover that 2x + y must be a multiple of 11, it's easier to build up all of the combinations of x and y that must be divisible by 11. So, for example, you could instead do:
3x + 7y is divisible by 11
14x + 7y is divisible by 11 (add 11x)
2x + y is divisible by 11 (factor out 7)
4x + 2y is divisible by 11 (multiply by 2)
4x - 9y is divisible by 11 (subtract 11y)
In any case, it doesn't seem much like any real GMAT question I've ever seen.
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