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# If both x and y are positive integers less than 100 and

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Manager
Joined: 24 Aug 2012
Posts: 104
If both x and y are positive integers less than 100 and  [#permalink]

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06 Nov 2012, 18:31
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Difficulty:

35% (medium)

Question Stats:

74% (01:43) correct 26% (01:42) wrong based on 180 sessions

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If both x and y are positive integers less than 100 and greater than 10, is the sum x + y a multiple of 11?

(1) x - y is a multiple of 22
(2) The tens digit and the units digit of x are the same; the tens digit and the units digit of y are the same.

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Re: If both x and y are positive integers less than 100 and  [#permalink]

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07 Nov 2012, 03:28
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If both x and y are positive integers less than 100 and greater than 10, is the sum x + y a multiple of 11?

(1) x - y is a multiple of 22. If x=33 and y=11, then the answer is YES but if x=34 and y=12, then the answer is NO. Not sufficient.

(2) The tens digit and the units digit of x are the same; the tens digit and the units digit of y are the same. Note that any two-digit integer can be represented as 10a+b (wher a and b are single digit integers), for example 37=3*10+7, 88=8*10+8, etc. Thus, we are given that x=10a+a=11a and y=10b+b=11b --> x+y=11a+11b=11(a+b). Therefore x+y is a multiple of 11. Sufficient.

OR: from (2) we have that both x and y must be multiples of 11 (11, 22, 33, 44, ..., 99). The sum of two multiples of 11 will give a multiple of 11. Sufficient.,

GENERALLY:
If integers $$a$$ and $$b$$ are both multiples of some integer $$k>1$$ (divisible by $$k$$), then their sum and difference will also be a multiple of $$k$$ (divisible by $$k$$):
Example: $$a=6$$ and $$b=9$$, both divisible by 3 ---> $$a+b=15$$ and $$a-b=-3$$, again both divisible by 3.

If out of integers $$a$$ and $$b$$ one is a multiple of some integer $$k>1$$ and another is not, then their sum and difference will NOT be a multiple of $$k$$ (divisible by $$k$$):
Example: $$a=6$$, divisible by 3 and $$b=5$$, not divisible by 3 ---> $$a+b=11$$ and $$a-b=1$$, neither is divisible by 3.

If integers $$a$$ and $$b$$ both are NOT multiples of some integer $$k>1$$ (divisible by $$k$$), then their sum and difference may or may not be a multiple of $$k$$ (divisible by $$k$$):
Example: $$a=5$$ and $$b=4$$, neither is divisible by 3 ---> $$a+b=9$$, is divisible by 3 and $$a-b=1$$, is not divisible by 3;
OR: $$a=6$$ and $$b=3$$, neither is divisible by 5 ---> $$a+b=9$$ and $$a-b=3$$, neither is divisible by 5;
OR: $$a=2$$ and $$b=2$$, neither is divisible by 4 ---> $$a+b=4$$ and $$a-b=0$$, both are divisible by 4.

Hope helps.
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Re: If both x and y are positive integers less than 100 and  [#permalink]

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26 Dec 2018, 08:30
kingb wrote:
If both x and y are positive integers less than 100 and greater than 10, is the sum x + y a multiple of 11?

(1) x - y is a multiple of 22
(2) The tens digit and the units digit of x are the same; the tens digit and the units digit of y are the same.

Given: x & y => +ive Integers between 10< x,y < 100

Q: is the sum x + y a multiple of 11

(1) x = 88 and y = 22 => (x-y)/ 22 is satisfied, answering the question as Yes.

Now when x = 64 and y = 20 => (x-y)/ 22 is satisfied, but when you answer the question x+y = 84/11, it will be a No.

1) will be not sufficient

(2) The tens digit and the units digit of x are the same; the tens digit and the units digit of y are the same

this means that x can be 11,22,33,44,55 and y can be 11,22,33,44,55

Now when you answer the question, it will be a consistent Yes.

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Re: If both x and y are positive integers less than 100 and   [#permalink] 26 Dec 2018, 08:30
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