Problem Solving

If y is an integer, then the least possible value of |23 - 5y| is
A) 1
B) 2
C) 3
D) 4
E) 5

To minimize |23-5y|, we need to find the value of y such that 23-5y is as close to zero as possible.

Try some values.
y=3: |23-5y| = |23-15| = |8| = 8
y=4: |23-5y| = |23-20| = |3| = 3
y=5: |23-5y| = |23-25| = |-2| = 2
y=6: |23-5y| = |23-30| = |-7| = 7

The least possible value of |23-5y| is 2.
Answer: B

Cheers,
Brent

GMATGuruNY wrote:

We can just test each answer choice:

A) 1. So, |23 - 5y| = |23 - 5(1)| = 18
B) 2. So, |23 - 5y| = |23 - 5(2)| = 13
C) 3. So, |23 - 5y| = |23 - 5(3)| = 8
D) 4. So, |23 - 5y| = |23 - 5(4)| = 3
E) 5. So, |23 - 5y| = |23 - 5(5)| = 2

So, when y = 5, |23 - 5y| = |23 - 5(5)| = |-2| = 2 (and 2 is the smallest possible value of |23 - 5y|)

The question asks, "What is the least possible value of |23 - 5y|?
So, the least possible value of |23 - 5y| is 2 (and this occurs when y = 5)

Can you please clarify the line of reasoning?
The answer choices represent the value of |23-5y|, but in the solution above they are being substituted for y.
If the answer choices were 0, 1, 2, 3, 4, this approach would seem to proceed as follows:

A) 0, So, |23 - 5y| = |23 - 5(0)| = 23
B) 1. So, |23 - 5y| = |23 - 5(1)| = 18
C) 2. So, |23 - 5y| = |23 - 5(2)| = 13
D) 3. So, |23 - 5y| = |23 - 5(3)| = 8
E) 4. So, |23 - 5y| = |23 - 5(4)| = 3

So, when y = 4, |23 - 5y| = |23 - 5(4)| = |3| = 3 (and 3 is the smallest possible value of |23 - 5y|).

However the least possible value of |23-5y| is not 3 but 2.

YIKES!
I was plugging in the answer choices as though they were the y-values!
Dumb, dumb, dumb!

I have edited my answer and am going back to bed!

Cheers,
Brent

rsarashi wrote:If y is an integer, then the least possible value of |23 - 5y| is

A) 1

B) 2

C) 3

D) 4

E) 5

OAB

Another approach is to check each answer choice to see if it COULD be the smallest possible value of |23 - 5y|

Let's start with answer choice A, since it is the smallest answer.

A) 1
Is it possible that |23 - 5y| = 1 if y MUST BE AN INTEGER?
Let's solve it.
If |23 - 5y| = 1, then 23 - 5y = 1 or 23 - 5y = -1

Take 23 - 5y = 1 and subtract 23 from both sides to get: -5y = -22
Solve to get: y = 4.4 NOT an integer

Take 23 - 5y = -1 and subtract 23 from both sides to get: -5y = -24
Solve to get: y = 4.8 NOT an integer

So, if y is an INTEGER, it's IMPOSSIBLE for |23 - 5y| to equal 1
ELIMINATE A


B) 2
Is it possible that |23 - 5y| = 2 if y MUST BE AN INTEGER?
Let's solve it.
If |23 - 5y| = 2, then 23 - 5y = 2 or 23 - 5y = -2

Take 23 - 5y = 2 and subtract 23 from both sides to get: -5y = -21
Solve to get: y = 4.2 NOT an integer

Take 23 - 5y = -2 and subtract 23 from both sides to get: -5y = -25
Solve to get: y = 5 AN INTEGER

AHA! It IS POSSIBLE for |23 - 5y| to equal 2

Answer: B

rsarashi wrote:If y is an integer, then the least possible value of |23 - 5y| is

A) 1

B) 2

C) 3

D) 4

E) 5

To solve this question, we must make sure we interpret it correctly. We are not finding the least possible value of y, but rather the least possible value of |23-5y| (the absolute value of 23 - 5y). Remember that the smallest value that can result from taking the absolute value is zero. Thus we need to make 23 - 5y as close to zero as possible.

We know that 5y is a multiple of 5, so let's first look at the multiples of 5 closest to 23. We have "20" and "25". Let's subtract both of these from 23 and see which one produces the smallest result. When 5y = 20, y is 4 and when 5y = 25, y is 5. Let's start with letting y = 4.

|23-5(4)|

|23-20|

|3| = 3

Next, let's let y equal 5.

|23-5(5)|

|23-25|

|-2| = 2

We see that the smallest possible value of |23-5y| is 2.

Answer: B

rsarashi wrote:If y is an integer, then the least possible value of |23 - 5y| is

A) 1

B) 2

C) 3

D) 4

E) 5

OAB

The least possible value of a number in modulus is 0.

Thus, the least possible value of |23 - 5y| = 0 => y = 23/5 = 4.6.

y = 4.6 is not possible since y is an integer, thus y can be either 4 or 5. Since '4.6' is relatively closer to '5' than to '4', |23 - 5y| would be least @y=5.

@y=4, |23 - 5y| = |23 - 5*4| = 3
@y=5, |23 - 5y| = |23 - 5*5| = 2 (Least possible value).

The correct answer: B

Relevant book: Manhattan Review GMAT Number Properties Guide

Hope this helps!

-Jay
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Hi rsarashi,

When a question involves basic arithmetic, sometimes the fastest way to get to the correct answer is just to use 'brute force' (and it's important to think in those terms - however you choose to approach a question, was "your way" the "fast way?") If you have a pacing issue, then you need to consider how you're handling ALL the questions - including the ones that you answered correctly.

Here, we're told that Y is an INTEGER and we're asked for the LEAST possible value of |23 - 5Y|. Be honest - how long would it really take you to plug in increasing integer values of Y until you found the LEAST value for that inequality? 15 seconds? 20 seconds? So put the pen on the pad and get to work...

Y = 1.... |23 - 5| = 18
Y = 2.... |23 - 10| = 13
Y = 3.... |23 - 15| = 8
Y = 4.... |23 - 20| = 3
Y = 5.... |23 - 25| = 2
Y = 6.... |23 - 30| = 7

Once you increase Y past Y = 5, the value of the absolute value increases, so we can stop working. The least value is 2.

Final Answer: B

GMAT assassins aren't born, they're made,
Rich