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When positive integer x is divided by 44

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When positive integer x is divided by 44

by Brent@GMATPrepNow » Thu Mar 30, 2017 7:34 am
When positive integer x is divided by 44, the remainder is 28. When positive integer y is divided by 22, the remainder is 14. If N = x+y, what is the remainder when N is divided by 11?

A) 1
B) 3
C) 5
D) 7
E) 9

Answer: E

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by Shameilia » Thu Mar 30, 2017 7:40 am
x can equal 28 and y can equal 14. 28 + 14 = 42. 42 divided by 11 gives a remainder of 9.
Answer is E.
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by Brent@GMATPrepNow » Thu Mar 30, 2017 7:52 am
Shameilia wrote:x can equal 28 and y can equal 14. 28 + 14 = 42. 42 divided by 11 gives a remainder of 9.
Answer is E.
Perfect!
Many students overlook the fact that we can simply find an x-value and a y-value that satisfy the given information, and then use those values to determine the remainder when x+y is divided by 11.

So, x COULD equal 28, and y COULD equal 14, which means N = 28 + 14 = 42
When we divide 42 by 11, the remainder is 9.

There's also an algebraic solution.
Anyone?

Cheers,
Brent
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by regor60 » Thu Mar 30, 2017 8:30 am
Brent@GMATPrepNow wrote:
There's also an algebraic solution.
Anyone?
Oh, all right.

X= 44N + 28 > first prompt

Y= 22M + 14 > second prompt

N=X+Y > N/11 = (X+Y)/11 = (44N + 22M + 42)/11

44N and 22M yield no remainders when divided by 11, only 42

42 = 3 + [spoiler]9/11[/spoiler] Remainder is 9
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by Brent@GMATPrepNow » Thu Mar 30, 2017 8:49 am
regor60 wrote:
Brent@GMATPrepNow wrote:
There's also an algebraic solution.
Anyone?
Oh, all right.

X= 44N + 28 > first prompt

Y= 22M + 14 > second prompt

N=X+Y > N/11 = (X+Y)/11 = (44N + 22M + 42)/11

44N and 22M yield no remainders when divided by 11, only 42

42 = 3 + [spoiler]9/11[/spoiler] Remainder is 9
Beautiful!!
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by Brent@GMATPrepNow » Thu Mar 30, 2017 8:55 am
Brent@GMATPrepNow wrote:When positive integer x is divided by 44, the remainder is 28. When positive integer y is divided by 22, the remainder is 14. If N = x+y, what is the remainder when N is divided by 11?

A) 1
B) 3
C) 5
D) 7
E) 9
Regor60's algebraic solution (above) is perfect.
In case some viewers are unfamiliar with this technique, here's a longer version:

When it comes to remainder questions, there's a useful rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

When x is divided by 44, the remainder is 28
We can rewrite this as "x divided by 44 equals some integer k with remainder 28"
So, x = 44k + 28 for some integer k

When integer y is divided by 22, the remainder is 14.
So, y = 22j + 14 for some integer j

N = x + y = (44k + 28) + (22j + 14)
N = 44k + 22j + 42
N = 44k + 22j + 33 + 9
N = 11(4k + 2j + 3) + 9

As we can see N is 9 GREATER than some multiple of 11
So, when we divide N by 11, the remainder will be 9

Answer: E
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