Is the positive two-digit integer N less than 40 ?

Is the positive two-digit integer N less than 40 ?

(1) The units digit of N is 6 more than the tens digit --> the greatest possible value of the units digit is 9, thus the greatest possible value of the tens digit is 9-6=3, which means that N is less than 40. Sufficient.

(2) N is 4 less than 4 times the units digit. The same here: the greatest possible value of the units digit is 9, thus the greatest possible value of N is 4*9-4=32. Sufficient.

Answer: D.
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When considering statement 2, isn't 28 the only correct number?

When units digit is 8,
\((4*units)-4 = (4*8)-4 = 32 - 4 = 28\)

The units digit of N=28 is also 8. So technically, N=24 or N=32 do not even satisfy the condition. But for DS, I guess (4*9)-4 is the fastest and best way.

I solved it using method of substitution.

Given number(N) is 2 digit number. So, 0<N<100. So, Question is whether N<40

Case 1:
If Number N is AB,
AB => A (A+6).
AB => 17,28,39.
hence, For sure, Number is less than 40.
Option is sufficient to answer the question.

As Option A is sufficient to answer the question, Answer Choices C, E are eliminated.

Case 2:
N is 4 less than 4 times the units digit.
Let's N is AB
N=> 4*B-4 => 4*(B-1)
So, B should be greater than 5.
If B =5, N =4 (Invalid case)
If B =6, N =8 (Invalid case)
IF B =7, N =24
If B =8, N =28
If B =9, N =32.
So, For sure Number is less than 40. Option is sufficient to answer the question.

So, Either statement is sufficient to answer this question.

You are right: 28 is the only two-digit number which satisfies the second statement.
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D.

1) let N = ab
=> b = a+6
b can maximum be 9, in which case a would be 3 (N = 39 max)
so N < 40 - Yes.
sufficient.

2) let N = ab
=> 10a+b - 4 = 4b
=> 10a-3b = 4
(a,b) = (1,2) is the only valid solution
so N < 40 - Yes
sufficient.
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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Is the positive two-digit integer N less than 40 ?

(1) The units digit of N is 6 more than the tens digit.
(2) N is 4 less than 4 times the units digit.

There are 3 variables (n,a,b) and 1 equation (n=10a+b) in the original condition, 2 equations in the given conditions, making it likely that (C) will be our answer.
Looking at the conditions,
For 1, n=17,28,39<40 answers the question yes, and is therefore sufficient.
For 2, 10a+b=4b-4, 10a+4=3b, and this answers the question 'yes' for n=28<40. This is sufficient as well, making the answer seem like (D).
However, this is a logic question with commonly made mistake type 4(B).

Hence, (C) also becomes the answer but if we apply integer commonly made mistake type 4, the answer still becomes (D).

For cases where we need 2 more equation, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Target question: Is N less than 40

Given: N is a positive two-digit integer

Statement 1: The units digit of N is 6 more than the tens digit
This statement is, essentially, restricting the value of the tens digit.
If the units digit is 6 more than the tens digit, then the tens digit cannot be very big.
For example, the tens digit cannot be 8, because the units digit would have to be 14, which is impossible.
Likewise, the tens digit cannot be 4, because the units digit would have to be 10, which is also impossible.
So, the greatest possible value of the tens digit of N is 3.
As such, N must be less than 40
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: N is 4 less than 4 times the units digit.
Well, 9 is the greatest possible value of a digit, and if the units digit were 9, then N would equal (4)(9) - 4, which is less than 40
So, no matter what value the units digit has, the resulting number (N), must be less than 40
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent
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We need to determine whether the positive two-digit integer N is less than 40.

Statement One Alone:

The units digit of N is 6 more than the tens digit.

With the information in statement one, we know the units digit is the larger of the two digits in N. So let’s say the units digit is 9 (the largest digit possible); then the tens digit will be 3 (since 9 is 6 more than 3). Thus this makes N = 39, which is less than 40. Now let’s say the units digit is 8; then the tens digit will be 2 and thus N = 28, which is still less than 40. Let’s say the units digit is 7; then the tens digit will be 1 and thus N = 17, which is again less than 40. At this point, we can’t make the units digits any smaller; if we did, the tens digits would be 0 or negative, but we know N is a positive two-digit integer.

Statement one is sufficient to answer the question.

Statement Two Alone:

N is 4 less than 4 times the units digit.

Again, let’s test some possible numerical values for the units digit. Let’s start with 9 again since it’s the largest digit possible. If the units digit is 9, then N = 4(9) – 4 = 32, which is less than 40. If the units digit is any smaller, then N will be less than 32, which means N will always be less than 40. If you can’t see this, look at the following:

If the units digit is 8, then N = 4(8) – 4 = 28.

If the units digit is 7, then N = 4(7) – 4 = 24, etc.

Statement two is also sufficient to answer the question.

Answer: D
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Could you help me by combining both statements, please?
Thanks__
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Sure.

To elaborate more: the question does not ask: what is the greatest possible value of N. The question asks: is N less than 40. From both statements we get an YES answer for the question, because even the greatest possible value of N from each statement is less than 40.

If you combine the statements, N turns out to be 28.

From (1) N could be: 39, 28, or 17.
From (2) N could be only 28.
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Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

Remember the relation between the Variable Approach, and Common Mistake Types 3 and 4 (A and B)[Watch lessons on our website to master these approaches and tips]

Step 1: Apply Variable Approach(VA)

Step II: After applying VA, if C is the answer, check whether the question is key questions.

StepIII: If the question is not a key question, choose C as the probable answer, but if the question is a key question, apply CMT 3 and 4 (A or B).

Step IV: If CMT3 or 4 (A or B) is applied, choose either A, B, or D.

Let's apply CMT (2), which says there should be only one answer for the condition to be sufficient. Also, this is an integer question and, therefore, we will have to apply CMT 3 and 4 (A or B).

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find Is the positive two-digit number N > 40?

=> Let the ten's digit be 'a' and unit's digit be 'b': N = 10a + b < 0 ?

Second and the third step of Variable Approach: From the original condition, we have 2 variables (a and b). To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 2 equations, C would most likely be the answer.

But we know that this is a key question [Integer question] and if we get an easy C as an answer, we will choose A or B.

Let’s take a look at each condition.

Condition(1) tells us that the units digit of N(b) is 6 more than the tens(a) digit.

=> b = a + 6 [possible combinations: a = 1, b = 7, a = 2, b = 8, a = 3, b = 9]

=> For maximum value of a= 3 and b = 9, N = 10(3) + 9 = 39 < 40 - YES

Since the answer is a unique YES , the condition(1) is alone sufficient by CMT 1.


Condition(2) tells us that N is 4 less than 4 times the units digit.

=> 10a + b = 4b - 4

=> For maximum value of b = 9, 10a + b = 4(9) - 4 = 32 < 40 - YES

Since the answer is unique, the condition(2) alone is sufficient by CMT 1.


Each condition alone is sufficient.

So, D is the correct answer.

Answer: D
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Answer: Option D

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Dear ScottTargetTestPrep, MathRevolution, Bunuel

all of you have written that in 2nd statement the solution could be N is 4*9-4=32. Sufficient.

Could you elaborate where am I wrong?

Statement (2) N is 4 less than 4 times the units digit.

N = A10 + B
A10 + B = 4B - 4 = > A10 - 3B = -4
So, to get negative 4 the last number ought to be greater the first number, and, thus, I only found the numbers A = 2 and B = 8. Thus, N = 28.

In your case when we are parsing 32 we have:
10A -3B = -4
10*3 - 3*2 = 30-6 = 24 that is not equal to - 4.

I know that you get 32 from the equation N = 4(9) – 4 = 32, then where am I wrong?
Why I could not get the same result 32?

Thank you in advance.

(1) The units digit of N is 6 more than the tens digit.
The maximum possibility is 39
Clearly sufficent

(2) N is 4 less than 4 times the units digit.
The only possibility that satifies the same = 28

Clearly sufficent

Therefore IMO D

EMPOWERgmatRichC
Dear Rich,
could you outline the error in my approach?
https://gmatclub.com/forum/is-the-posit ... l#p2877231

Thanks beforehand.

Hi BLTN,

Based on the information in Fact 2, the units digit of N CANNOT be 9. Here's why:

IF... the units digit of N was 9, then "4 less than 4 times the units digit would be.... 4(9) - 4 = 36 - 4 = 32.... meaning that N = 32.... but 32 has a units digit of '2' (and in this example, it's supposed to be a '9'). Thus, this is NOT a valid example that fits the information in Fact 2.

That having been said, what this example DOES prove is that whatever 2-digit number that N could be, that option (or options) MUST be LESS than 32 (since the largest possible units digit is '9' - and by extension, the largest possible value of N cannot be any greater than the resulting 32). Notice how the question asks "if... N is less than 40?" This bit of work - even though it technically does not 'fit' the information in Fact 2 - does provide enough evidence that N will ALWAYS be less than 40 (and more specifically, N will ALWAYS be less than 32).

GMAT assassins aren't born, they're made,
Rich
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Hi Bunuel, I get the explanation but I am still a little confused.
Statement 1 says that unit digit of N is 6 more than tens digit. I can't stop thinking about this statement. According to my translation, If the greates vale of units digit is 9 then the tens digit is 6 less than the units digit so 97
Similarly, if unit digit of N is 8 then tens digit will be 2 there for N is 82.

Statement 2 makes sense.

What am i getting wrong?

Say N is ab, where a is the tens digit and b is the units digit.

(1) says that "1) The units digit of N is 6 more than the tens digit ", so the greatest possible value of the units digit, which is b, is 9, thus the greatest possible value of the tens digit, which is a, is 9-6=3, which means that N is less than 40. Sufficient.

(2) says that "N is 4 less than 4 times the units digit". The same here: the greatest possible value of the units digit, which is b, is 9, thus the greatest possible value of N is 4*9-4=32. Sufficient.

Hope it's clear.
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Hi,
I apply the same logic, but why did you write "10a+b - 4 = 4b"
I think it is: 10a+b= 4b -4 since N=4b - 4