Embed Size (px)
Citation preview
GMAT QUANTITATIVE REASONING
NUMBER PROPERTIES
REMAINDERS
DATA SUFFICIENCY
QUESTION 5
Q-51 Series
Question
What is the remainder when the positive integer x is divided
by 6?
1. When x is divided by 7, the remainder is 5.
2. When x is divided by 9, the remainder is 3.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
◴Answer these questions before evaluating the statements.
Determine approach to solve the question
Step 1
Spend a few seconds answering the following questions
Before going to the statements
When is the data sufficient?
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
Spend a few seconds answering the following questions
Before going to the statements
When is the data sufficient?
The data is sufficient when we are able to find a uniqueremainder.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
Spend a few seconds answering the following questions
Before going to the statements
When is the data sufficient? When is it not sufficient?
The data is sufficient when we are able to find a uniqueremainder.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
Spend a few seconds answering the following questions
Before going to the statements
When is the data sufficient? When is it not sufficient?
The data is sufficient when we are able to find a uniqueremainder.
The data is NOT sufficient when we get more than one remainder using the information in the statements
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
Spend a few seconds answering the following questions
Before going to the statements
When is the data sufficient? When is it not sufficient? What do we know about x?
The data is sufficient when we are able to find a uniqueremainder.
The data is NOT sufficient when we get more than one remainder using the information in the statements
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
Spend a few seconds answering the following questions
Before going to the statements
When is the data sufficient? When is it not sufficient? What do we know about x?
The data is sufficient when we are able to find a uniqueremainder.
The data is NOT sufficient when we get more than one remainder using the information in the statements
x is a positive integer
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
Look for a counter example
Approach to solve the question
Look for a counter example
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
Look for a counter example
Approach to solve the question
Look for a counter example
If a counter example exists
data NOT sufficient
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
Look for a counter example
Approach to solve the question
Look for a counter example
If a counter example exists
data NOT sufficient
If a counter example does not
exist data sufficient
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
Look for a counter example
Approach to solve the question
Look for a counter example
If a counter example exists
data NOT sufficient
If a counter example does not
exist data sufficient
What is a counter example?
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
Look for a counter example
Approach to solve the question
Look for a counter example
If a counter example exists
data NOT sufficient
If a counter example does not
exist data sufficient
What is a counter example?
Find two values for x satisfying information in the
statement, each one resulting in a different
remainder when divided by 6.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
Look for a counter example
Approach to solve the question
Look for a counter example
If a counter example exists
data NOT sufficient
If a counter example does not
exist data sufficient
What is a counter example?
If two such values exist then a counter example
exists
Find two values for x satisfying information in the
statement, each one resulting in a different
remainder when divided by 6.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
Look for a counter example
Approach to solve the question
Look for a counter example
If a counter example exists
data NOT sufficient
If a counter example does not
exist data sufficient
What is a counter example?
If two such values exist then a counter example
exists
Find two values for x satisfying information in the
statement, each one resulting in a different
remainder when divided by 6.
If for all x satisfying information in the statement,
the remainder is unique – a counter example does
not exist.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
◴Evaluate Statement 1 alone
Step 2
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 1 : When x is divided by 7, the remainder is 5.
Yes
Look for a counter example
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 1 : When x is divided by 7, the remainder is 5.
Yes
Look for a counter example
1
x = 5
Leaves a remainder
of 5 when divided
by 7.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 1 : When x is divided by 7, the remainder is 5.
Yes
Look for a counter example
1
x = 5
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 6.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 1 : When x is divided by 7, the remainder is 5.
Yes
Look for a counter example
1
x = 5
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 6.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
How do we find the next number that satisfies statement 1?
Values satisfying first statement will be in an Arithmetic progression. First term is 5 and common difference is 7. The sequence is 5, 12, 19, 26 …..
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 1 : When x is divided by 7, the remainder is 5.
Yes
Look for a counter example
1
x = 5
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 6.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
How do we find the next number that satisfies statement 1?
Values satisfying first statement will be in an Arithmetic progression. First term is 5 and common difference is 7. The sequence is 5, 12, 19, 26 …..
Lets find out the remainder when x = 12
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 1 : When x is divided by 7, the remainder is 5.
Yes
Look for a counter example
1 2
x = 5
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 6.
x = 12
Leaves a remainder
of 5 when divided
by 7.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 1 : When x is divided by 7, the remainder is 5.
Yes
Look for a counter example
1 2
x = 5
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 6.
x = 12
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 0 when divided
by 6.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 1 : When x is divided by 7, the remainder is 5.
Yes
Look for a counter example
1 2
x = 5
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 6.
x = 12
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 0 when divided
by 6.
Counter example exists
Statement 1 alone is NOT sufficient
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 1 : When x is divided by 7, the remainder is 5.
Yes
Look for a counter example
1 2
x = 5
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 6.
x = 12
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 0 when divided
by 6.
Eliminate choices A and D
Counter example exists
Statement 1 alone is NOT sufficient
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 1 : When x is divided by 7, the remainder is 5.
Yes
Look for a counter example
1 2
x = 5
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 6.
x = 12
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 0 when divided
by 6.
Choices narrow down to B, C or E.
Eliminate choices A and D
Counter example exists
Statement 1 alone is NOT sufficient
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
◴Evaluate Statement 2 alone
Step 3
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 2 : When x is divided by 9, the remainder is 3.
Yes
Look for a counter example
1
x = 3
Leaves a remainder
of 3 when divided
by 9.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 2 : When x is divided by 9, the remainder is 3.
Yes
Look for a counter example
1
x = 3
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 3 when divided
by 6.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 2 : When x is divided by 9, the remainder is 3.
Yes
Look for a counter example
1
x = 3
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 3 when divided
by 6.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
How do we find the next number that satisfies statement 2?
Values satisfying second statement will be in an Arithmetic progression. First term is 3 and common difference is 9. The sequence is 3, 12, 21, 30 …..
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 2 : When x is divided by 9, the remainder is 3.
Yes
Look for a counter example
1
x = 3
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 3 when divided
by 6.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
How do we find the next number that satisfies statement 2?
Values satisfying second statement will be in an Arithmetic progression. First term is 3 and common difference is 9. The sequence is 3, 12, 21, 30 …..
Lets find out the remainder when x = 12
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 2 : When x is divided by 9, the remainder is 3.
Yes
Look for a counter example
1 2
x = 3
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 3 when divided
by 6.
x = 12
Leaves a remainder
of 3 when divided
by 9.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 2 : When x is divided by 9, the remainder is 3.
Yes
Look for a counter example
1 2
x = 3
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 3 when divided
by 6.
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 2 : When x is divided by 9, the remainder is 3.
Yes
Look for a counter example
1 2
x = 3
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 3 when divided
by 6.
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
Counter example exists
Statement 2 alone is NOT sufficient
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 2 : When x is divided by 9, the remainder is 3.
Yes
Look for a counter example
1 2
x = 3
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 3 when divided
by 6.
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
Eliminate choice B as well
Counter example exists
Statement 2 alone is NOT sufficient
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
Lets evaluate statement 2 : When x is divided by 9, the remainder is 3.
Yes
Look for a counter example
1 2
x = 3
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 3 when divided
by 6.
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
Choices narrow down to C or E.
Eliminate choice B as well
Counter example exists
Statement 2 alone is NOT sufficient
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
◴Evaluate the data given in the two statements together
Step 4
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
x = 12
Leaves a remainder
of 5 when divided
by 7.
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 5 when divided
by 7.
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
Leaves a remainder
of 5 when divided
by 7.
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
How do we find the next number that satisfies both the statements?
Leaves a remainder
of 5 when divided
by 7.
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
How do we find the next number that satisfies both the statements?
Leaves a remainder
of 5 when divided
by 7.
Values satisfying first statement are in an Arithmetic progression. First term is 5 and common difference is 7. The sequence is 5, 12, 19, 26 …..
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
How do we find the next number that satisfies both the statements?
Leaves a remainder
of 5 when divided
by 7.
Values satisfying first statement are in an Arithmetic progression. First term is 5 and common difference is 7. The sequence is 5, 12, 19, 26 …..
Values satisfying second statement are in a second Arithmetic progression. First term is 3 and common difference is 9. The sequence is 3, 12, 21, 30 …..
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
How do we find the next number that satisfies both the statements?
Leaves a remainder
of 5 when divided
by 7.
Values satisfying first statement are in an Arithmetic progression. First term is 5 and common difference is 7. The sequence is 5, 12, 19, 26 …..
Values satisfying second statement are in a second Arithmetic progression. First term is 3 and common difference is 9. The sequence is 3, 12, 21, 30 …..
Therefore, values common to both statements will be in an AP. First term will be 12. Common difference will be the LCM of the two common differences. i.e., LCM of 7 and 9 = 63. The sequence will be 12, 75, 138 …
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
x = 75
Leaves a remainder
of 5 when divided
by 7.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
x = 75
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 7.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
x = 75
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 7.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
x = 75
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 3 when divided
by 6.
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 7.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
x = 75
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 3 when divided
by 6.
Counter example exists
Statements Together are NOT sufficient
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 7.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
x = 75
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 3 when divided
by 6.
Eliminate choice C as well
Counter example exists
Statements Together are NOT sufficient
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 7.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
What is the remainder when the positive integer x is divided by 6?
x is divided by 7, the remainder is 5. x is divided by 9, the remainder is 3.
Yes
Look for a counter example
x = 12
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 0 when divided
by 6.
x = 75
Leaves a remainder
of 3 when divided
by 9.
Leaves a remainder
of 3 when divided
by 6.Answer choice E.
Eliminate choice C as well
Counter example exists
Statements Together are NOT sufficient
Leaves a remainder
of 5 when divided
by 7.
Leaves a remainder
of 5 when divided
by 7.
AnswerCombineStatement 2Statement 1ApproachQuestion StemQuestion
More Hard Math Questions
Visit www.q-51.com
Queries, feedback?
Reach us at [email protected]
GMAT Classes @ Chennai, India
Weekend and weekday GMAT classes by US B School graduates, GMAT 98%lers.
Chennai.4gmat.com or +91 95000 48484
GMAT Classes @ Bangalore, India
Weekend classes by US B school graduates.
Bangalore.4gmat.com or +91 74060 48484
