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Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
If x and y are integers greater than 3, and 15y – 11x = 8
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11 Sep 2019, 09:16
11 Sep 2019, 09:16
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Question Stats:
57% (03:17) correct
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If x and y are integers greater than 3, and 15y – 11x = 8, what is the least possible value of x+y?
A) 12
B) 16
C) 26
D) 30
E) 34
A) 12
B) 16
C) 26
D) 30
E) 34
Re: If x and y are integers greater than 3, and 15y – 11x = 8
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11 Sep 2019, 10:25
11 Sep 2019, 10:25
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GreenlightTestPrep wrote:
If x and y are integers greater than 3, and 15y – 11x = 8, what is the least possible value of x+y?
A) 12
B) 16
C) 26
D) 30
E) 34
A) 12
B) 16
C) 26
D) 30
E) 34
y = (8 +11x)/15
since y is an integer we're looking for when 15 is a factor of 8 + 11x. It happens at 2 but this isn't larger than 3. To come up with some potential x's, I just examine the 1s digit and looks for when I can get either a 5 or a 0.
numerator(x) = 8+11x
Potentials = { 7, 12, 17, 22}
numerator(7) = 85 which is NOT divisible by 15.
numerator(12) = 140 which is NOT divisible by 15.
numerator(17) = 8 + 170 + 17 = 195 which IS divisible by 15
195/15 = 13
now 17 + 13 = 30.
D.
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: If x and y are integers greater than 3, and 15y – 11x = 8
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11 Sep 2019, 13:57
11 Sep 2019, 13:57
1
GreenlightTestPrep wrote:
If x and y are integers greater than 3, and 15y – 11x = 8, what is the least possible value of x+y?
A) 12
B) 16
C) 26
D) 30
E) 34
A) 12
B) 16
C) 26
D) 30
E) 34
Given: 15y – 11x = 8
Subtract 4y from both sides to get: 11y – 11x = 8 – 4y
ASIDE: Why did I subtract 4y from both sides?
This allows me to factor the expression 11y – 11x, which may help reveal some useful relationship.
Continuing along….
Factor both sides to get: 11(y – x) = 4(2 – y)
KEY CONCEPT: Since x and y are INTEGERS, we know that (y – x) is an INTEGER, which means 11(y – x) is a multiple of 11
From this, we can conclude that 4(2 – y) is a multiple of 11
What is the smallest value of y (given that y is an integer greater than 3) such that 4(2 – y) is a multiple of 11?
If y=13, then 4(2 – y) = 4(2 – 13) = 4(-11) = -44. Perfect!
So, y=13 is the smallest value of y to meet the given conditions.
To find the corresponding value of x, take 15y – 11x = 8 and plug in y=13 to get: 15(13) – 11x = 8
Simplify : 195 – 11x = 8
Subtract 195 from both sides: -11x = -187
Solve: x = 17
So, the LEAST possible value of x+y = 17 + 13 = 30
Answer: D
Cheers,
Brent
