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QUESTION: 1

If both 11^{2} and 3^{3} are factors of the number a * 4^{3} * 6^{2} * 13^{11}, then what is the smallest possible value of 'a'?

Solution:

Step 1 of solving this GMAT Number Properties Question: Prime factorize the given expression

a * 4^{3} * 6^{2} * 13^{11} can be expressed in terms of its prime factors as a * 2^{8} * 3^{2} * 13^{11}

Step 2 of solving this GMAT Number Properties Question: Find factors missing after excluding 'a' to make the number divisible by both 11^{2} and 3^{3}

11^{2} is a factor of the given number.

If we do not include 'a', 11 is not a prime factor of the given number.

If 11^{2} is a factor of the number, 11^{2} should be a part of 'a'

3^{3} is a factor of the given number.

If we do not include 'a', the number has only 3^{2} in it.

Therefore, if 3^{3} has to be a factor of the given number 'a' has to contain 3^{1} in it.

Therefore, 'a' should be at least 11^{2} * 3 = 363 if the given number has 11^{2} and 3^{3} as its factors.

The question is **"what is the smallest possible value of 'a'?"**

The smallest value that 'a' can take is **363**

QUESTION: 2

How many different positive integers exist between 10^{6} and 10^{7}, the sum of whose digits is equal to 2?

Solution:

Method 1 to solve this GMAT Number Properties Question: Find the number of such integers existing for a lower power of 10 and extrapolate the results.

Between 10 and 100, that is 10^{1} and 10^{2}, we have 2 numbers, 11 and 20.

Between 100 and 1000, that is 10^{2} and 10^{3}, we have 3 numbers, 101, 110 and 200.

Therefore, between 10^{6} and 10^{7}, one will have 7 integers whose sum will be equal to 2.

Alternative approach

All numbers between 10^{6} and 10^{7} will be 7 digit numbers.

There are two possibilities if the sum of the digits has to be '2'.

**Possibility 1**: Two of the 7 digits are 1s and the remaining 5 are 0s.

The left most digit has to be one of the 1s. That leaves us with 6 places where the second 1 can appear.

So, a total of __six__ 7-digit numbers comprising two 1s exist, sum of whose digits is '2'.

**Possibility 2**: One digit is 2 and the remaining are 0s.

The only possibility is 2000000.

Total count is the sum of the counts from these two possibilities = 6 + 1 = 7

Choice B is the correct answer.

QUESTION: 3

A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?

Solution:

Step 1 of solving this GMAT Number Properties Question: Decode "A number when divided by a divisor leaves a remainder of 24"

Let the original number be 'a'.

Let the divisor be 'd'.

Let the quotient of dividing 'a' by 'd' be 'x'.

Therefore, we can write the division as a/d = x

and the remainder is 24.

i.e., a = dx + 24

Step 2 of solving this GMAT Number Properties Question: Decode "When twice the original number is divided by the same divisor, the remainder is 11"

Twice the original number is divided by 'd' means 2a is divided by d.

We know from Step 1 that a = dx + 24.

Therefore, 2a = 2(dx + 48) or 2a = 2dx + 48

When (2dx + 48) is divided by 'd' the remainder is 11.

2dx is divisible by 'd' and will therefore, not leave a remainder.

The remainder of 11 would be the remainder of dividing 48 by d.

The question essentially becomes **"What number will leave a remainder of 11 when it divides 48?"**

When 37 divides 48, the remainder is 11.

Hence, the divisor is 37.

Choice D is the correct answer.

QUESTION: 4

How many keystrokes are needed to type numbers from 1 to 1000?

Solution:

While typing numbers from 1 to 1000, there are 9 single digit numbers: from 1 to 9.

Each of these numbers requires one keystroke.

That is 9 key strokes.

There are 90 two-digit numbers: from 10 to 99.

Each of these numbers requires 2 keystrokes.

Therefore, 180 keystrokes to type the 2-digit numbers.

There are 900 three-digit numbers: from 100 to 999.

Each of these numbers requires 3 keystrokes.

Therefore, 2700 keystrokes to type the 3-digit numbers.

1000 is a four-digit number which requires 4 keystrokes.

Totally, therefore, one requires 9 + 180 + 2700 + 4 = 2893 keystrokes.

Choice B is the correct answer.

QUESTION: 5

When 242 is divided by a certain divisor the remainder obtained is 8. When 698 is divided by the same divisor the remainder obtained is 9. However, when the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is 4. What is the value of the divisor?

Solution:

When 242 is divided by a certain divisor the remainder obtained is 8.

Let the divisor be d.

When 242 is divided by d, let the quotient be 'x'. The remainder is 8.

Therefore, 242 = xd + 8

When 698 is divided by the same divisor the remainder obtained is 9.

Let y be the quotient when 698 is divided by d.

Then, 698 = yd + 9.

When the sum of the two numbers, 242 and 698, is divided by the divisor, the remainder obtained is 4.

242 + 698 = 940 = xd + yd + 8 + 9

940 = xd + yd + 17

Because xd and yd are divisible by d, the remainder when 940 is divided by d should have been 17.

However, because we know that the remainder is 4, it would be possible only when 17d17d leaves a remainder of 4.

If the remainder obtained is 4 when 17 is divided by 'd', then 'd' has to be 13.

Choice C is the correct answer.

QUESTION: 6

How many integral divisors does the number 120 have?

Solution:

Step 1 of solving this GMAT Number Properties Question: Express the number in terms of its prime factors

120 = 2^{3} * 3 * 5.

The three prime factors are 2, 3 and 5.

The powers of these prime factors are 3, 1 and 1 respectively.

Step 2 of solving this GMAT Number Properties Question:Find the number of factors as follows

To find the number of factors / integral divisors that 120 has, increment the powers of each of the prime factors by 1 and then multiply them.

**Number of factors = (3 + 1) * (1 + 1) * (1 + 1) = 4 * 2 * 2 =**16

Choice B is the correct answer.

QUESTION: 7

How many trailing zeros will be there after the rightmost non-zero digit in the value of 25!?

Solution:

5! means factorial 25 whose value = 25 * 24 * 23 * 22 *....* 1

When a number that is a multiple of 5 is multiplied with an even number, it results in a trailing zero.

(Product of 5 and 2 is 10 and any number when multiplied with 10 or a power of 10 will have one or as many zeroes as the power of 10 with which it has been multiplied)

In 25!, the following numbers have 5 as their factor: 5, 10, 15, 20, and 25.

25 is the square of 5 and hence it has two 5s in it.

In toto, it is equivalent of having six 5s.

There are at least 6 even numbers in 25!

Hence, the number 25! will have 6 trailing zeroes in it.

Choice C is the correct answer.

QUESTION: 8

What is the remainder when 1044 * 1047 * 1050 * 1053 is divided by 33?

Solution:

You can solve this problem if you know this rule about remainders.

Let a number x divide the product of A and B.

The remainder will be the product of the remainders when x divides A and when x divides B.

Using this rule,

The remainder when 33 divides 1044 is 21.

The remainder when 33 divides 1047 is 24.

The remainder when 33 divides 1050 is 27.

The remainder when 33 divides 1053 is 30.

∴ the remainder when 33 divides 1044 * 1047 * 1050 * 1053 is 21 * 24 * 27 * 30.

**Note:** The remainder when a number is divided by a divisor 'd' will take values from 0 to (d - 1). It will not be equal to or more than 'd'.

The value of 21 * 24 * 27 * 30 is more than 33.

When the value of the remainder is more than the divisor, the final remainder will be the remainder of dividing the product by the divisor.

i.e., the final remainder is the remainder when 33 divides 21 * 24 * 27 * 30.

When 33 divides 21 * 24 * 27 * 30, the remainder is 30.

Choice C is the correct answer.

QUESTION: 9

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

Q.

Is x^{3} > x^{2}?

1. x > 0

2. x < 1

__Numbers:__ All numbers used are real numbers.

Solution:

Step 1 of solving this GMAT DS question: Understand the question stem and when the data is sufficient

**What kind of an answer will the question fetch?**

The question is an **"Is"** question. Answer to an "is" question is either YES or NO.

**When is the data sufficient?**

The data is sufficient if we are able to get a DEFINITE YES or a DEFINITE NO from the information given in the statements.

If from the statements we get an answer that x^{3} > x^{2} in some instances and it is otherwise in other instances, the data is NOT sufficient.

Step 2 of solving this GMAT DS question: Evaluate Statement (1) ALONE

Statement 1: x > 0

We know that x is a positive number.

Interval 1: If 0 < x < 1, then x^{3} < x^{2}.

For example, (0.5)^{3} = 0.125, which is lesser than (0.5)^{2} = 0.25

The answer to the question is NO.

Interval 2: If x > 1, then x^{3} > x^{2}

For example, 2^{3} = 8 which is greater than 2^{2} = 4

The answer to the question is YES.

We do NOT have a DEFINITE answer using statement 1.

**Statement 1 ALONE is NOT sufficient.**

Eliminate choices A and D. Choices narrow down to B, C or E.

Step 3 of solving this GMAT DS question: Evaluate Statement (2) ALONE

Statement 2: x < 1

Interval 1: For positive values of x, i.e., 0 < x < 1, we know x^{3} < x^{2}.

The answer to the question is NO.

Interval 2:For negative values of x, x^{3} will be a negative number and x^{2} will be a positive number.

Hence, x^{3} < x^{2}

The answer to the question is NO.

**Lastly, what is the answer if x = 0?**

When x = 0, x^{3} = x^{2}.

The answer to the question is NO.

Hence, if we know that x < 1, we can conclude that x^{3} is __NOT GREATER THAN__ x^{2}.

We have a DEFINITE answer, even if it is NO.

Statement 2 ALONE is sufficient. Eliminate choices C and E.

Choice B is the answer.

QUESTION: 10

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

Q. Is x/y a terminating decimal?

1. x is a multiple of 2

2. y is a multiple of 3

Solution:

Step 1 of solving this GMAT DS question: Understand the Question Stem

What kind of an answer will the question fetch?

The question is an "Is" question. Answer to an "is" question is either YES or NO.

When is the data sufficient?

The data is sufficient if we are able to get a DEFINITE YES or a DEFINITE NO from the information given in the statements.

If the statements do not have adequate data to provide a CONCLUSIVE answer and we get YES for some instances and NO for others, the given data is NOT sufficient.

What is a terminating decimal?

Numbers that have finite number of digits after the decimal point are called terminating or finite decimals. e.g., 2.5, 0.25, 8.

Note: All integers are terminating decimals.

Numbers that have infinite digits after the decimal point are called non-terminating or infinite decimals

Step 2 of solving this GMAT DS question: Evaluate Statement (1) ALONE

Statement (1) : x is a multiple of 2

No information about y has been provided.

Approach: Let us look for a counter example.

Example: When x = 2 and y = 3, x/y is non-terminating.

The answer to the question is NO.

Counter Example: When x = 2 and y = 4, x/y is terminating.

The answer to the question is YES.

We have found a counter example. Therefore, statement 1 does not provide a DEFINITE answer.

**Statement 1 ALONE is NOT sufficient.**

Eliminate choices A and D. Choices narrow down to B, C or E.

Step 3 of solving this GMAT DS question: Evaluate Statement (2) ALONE

Statement (2) : y is a multiple of 3

No information about x has been provided.

Approach: Let us look for a counter example.

Example:When x = 3 and y = 3, xyxy is terminating.

The answer to the question is YES.

Counter Example: When x = 2 and y = 3, x/y is non-terminating.

The answer to the question is NO.

We have found a counter example. Therefore, statement 2 does not provide a DEFINITE answer.

**Statement 2 ALONE is NOT sufficient.**

Eliminate choice B. Choices narrow down to C or E.

Step 4 of solving this GMAT DS question Evaluate Statements (1) & (2) Together

Statements Together : x is a multiple of 2 & y is a multiple of 3

Approach: Let us look for a counter example.

Example:When x = 6 and y = 6, x/y is terminating.

The answer to the question is YES.

Counter Example: When x = 2 and y = 3, x/y is non-terminating.

The answer to the question is NO.

We have found a counter example. Despite combining the information in the statements we are not able to find a DEFINITE answer.

Statements TOGETHER are NOT SUFFICIENT. Choice E is the answer.

QUESTION: 11

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

Q. Is the positive integer X divisible by 21?

1. When X is divided by 14, the remainder is 4

2. When X is divided by 15, the remainder is 5

Solution:

Step 1 of solving this GMAT DS question: Understand the Question Stem

**What kind of an answer will the question fetch?**

The question is an **"Is"** question. Answer to an "is" question is either YES or NO.

**When is the data sufficient?**

The data is sufficient if we are able to get a DEFINITE YES or a DEFINITE NO from the information given in the statements.

If we get a MAYBE as an answer, the data is NOT sufficient

**Do we have any more information about 'X' from the question stem?**

The question stem states that 'X' is a positive integer.

**What kind of numbers will be divisible by 21?**

A number is divisible by 21 if it is divisible by 3 and 7.

Step 2 of solving this GMAT DS question Evaluating Statement (1) ALONE

Statement (1) : When X is divided by 14, the remainder is 4

The number is, therefore, of the form 14k + 4.

It will leave a remainder of 4 when divided by 7. (14k is divisible by 7. When 4 is divided by 7, the remainder is 4.)

This number is definitely not divisible by 7.

To be divisible by 21, the number must be divisible by both 3 and 7. This number is not divisible by 7. Hence, X is not divisible by 21.

We have a DEFINITE NO as the answer to the question using statement 1.

**Statement 1 ALONE is sufficient.**

Eliminate choices B, C and E. Choices narrow down to A or D.

Step 3 of solving this GMAT DS question Evaluating Statement (2) ALONE

Statement (2) : When X is divided by 15, the remainder is 5

The number X is of the form 15m + 5

Therefore, the number will leave a remainder of 2 when divided by 3.

Hence, it is not divisible by 3.

To be divisible by 21, the number must be divisible by both 3 and 7. This number is not divisible by 3. Hence, X is not divisible by 21.

We have a DEFINITE NO as the answer to the question using statement 2 as well.

**Statement 2 ALONE is also sufficient.**

Eliminate choice A.

QUESTION: 12

Q. If x and y are positive integers, is y odd?

1. x is odd.

2. xy is odd.

Solution:

Step 1 of solving this GMAT DS question: Understand the Question Stem

**What kind of an answer will the question fetch?**

The question is an **"Is"** question. Answer to an "is" question is either YES or NO.

**When is the data sufficient?**

The data is sufficient if we are able to get a DEFINITE YES or a DEFINITE NO from the information given in the statements.

If the answer is __MAYBE__ or __DONOT KNOW__, the data is NOT sufficient.

**Do we have any more information about x and y from the question stem?**

The question stem states that both x and y are positive integers.

Step 2 of solving this GMAT DS question: Evaluate Statement (1) ALONE

Statement (1) : x is odd.

The statement provides no information about y.

**Statement 1 ALONE is NOT sufficient.**

Eliminate choices A and D. Choices narrow down to B, C or E.

Step 3 of solving this GMAT DS question: Evaluate Statement (2) ALONE

Statement (2) : xy is odd.

Given that x and y are integers and that the product xy is odd, both x and y have to be odd (product of two odd integers is odd).

We can answer the question with a definite YES.

Eliminate choices C and E.

Statement 2 ALONE is sufficient. Choice B is the answer.

QUESTION: 13

Q. Is xy < 0?

1. 5|x| + |y| = 0

2. |x| + 5|y| = 0

Solution:

Step 1 of solving this GMAT DS question: Understand the Question Stem

**What kind of an answer will the question fetch?**

The question is an **"Is"** question. Answer to an "is" question is either YES or NO.

**When is the data sufficient?**

The data is sufficient if we are able to get a DEFINITE YES or a DEFINITE NO from the information given in the statements.

If the answer is __MAYBE__ or __DONOT KNOW__, the data is NOT sufficient.

**When is the answer YES and when is it NO?**

If xy < 0, the answer is YES. If xy ≥ 0, the answer is NO.

__Note__: If xy = 0, the answer is NO.

Step 2 of solving this GMAT DS question :

Evaluate Statement (1) ALONE: 5|x| + |y| = 0

Modulus of any real number will always be non-negative.

So, if 5|x| + |y| = 0, then the only possibility is that x = 0 and y = 0.

Therefore, xy = 0.

Or xy is not negative.

The answer is a DEFINITE NO.

**Statement 1 ALONE is sufficient.**

Eliminate choices B, C and E. Choices narrow down to A or D.

Step 3 of solving this GMAT DS question :

Evaluate Statement (2) ALONE: |x| + 5|y| = 0

Modulus of any real number will always be non-negative.

So, if |x| + 5|y| = 0, then the only possibility is that x = 0 and y = 0.

Therefore, xy = 0.

Or xy is not negative.

The answer is a DEFINITE NO.

**Statement 2 ALONE is also sufficient.**

Eliminate choice A.

Each statement is INDEPENDENTLY sufficient. Choice D is the answer.

QUESTION: 14

Q. When a positive integer 'x' is divided by a divisor 'd', the remainder is 24. What is d?

1. When 2x is divided by d, the remainder is 23.

2. When 3x is divided by d, the remainder is 22

Solution:

Step 1 of solving this GMAT DS question: Understand the Question Stem

**What kind of an answer will the question fetch?**

The question is a "What is the value?" question. For questions asking for a value, the answer should be a number.

**When is the data sufficient?**

The data is sufficient if we are able to get a UNIQUE answer for the value of 'd' from the information in the statements.

If either the statements do not have adequate data to determine the value of 'd' or if more than one value of 'd' exists based on the information in the statement, the data is NOT sufficient.

**What do we know from the question stem?**

'x' is a positive integer. Dividing x by d leaves a remainder of 24.

So, the value of 'd' is more than 24.

Step 2 of solving this GMAT DS question:

Evaluate Statement (1) ALONE: When 2x is divided by d, the remainder is 23.

The question stem states that when x is divided by d, the remainder is 24.

Therefore, when 2x is divided by d, the remainder should be 2 * 24 = 48.

However, from statement (1) we know that the remainder is 23. We can infer the following from the question stem and statement 1:

- the divisor d is less than 48
- the divisor is at least 25 and
- 48 divided by divisor d should leave a remainder of 23.

i.e., 48 = nd + 23 or nd = 25.

The possible values for d are 1, 5 and 25.

However, as d is at least 25, the divisor cannot be 1 or 5.

So, we can conclude that 25 is the divisor.

**Statement 1 ALONE is sufficient.**

Eliminate choices B, C and E. Choices narrow down to A or D.

Step 3 of solving this GMAT DS question:

Evaluate Statement (2) ALONE: When 3x is divided by d, the remainder is 22.

If x leaves a remainder of 24 when divided by d, then 3x will leave a remainder of 3 * 24 = 72 when divided by d.

However, the remainder is 22.

This tells us that the divisor is less than 72 and that 72 divided by d leaves a remainder of 22.

So, 72 = n * d + 22

Or nd = 72 - 22 = 50

If nd = 50, d could be 50 or 25 or 10 or 5 or 2.

However, from the question stem we have deduced that the divisor is at least 25. So, d cannot be 10, 5 and 2.

But, d could be 25 or 50.

From statement 2, we are unable to deduce a unique value for d.

**Statement 2 ALONE is NOT sufficient.**

Eliminate choice D.

Statement 1 ALONE is sufficient. Choice A is the answer.

QUESTION: 15

Q. How many of the numbers x, y, and z are positive if each of these numbers is less than 10?

1. x + y + z = 20

2. x + y = 14

Solution:

Step 1 of solving this GMAT DS question: Understand the Question Stem

**What kind of an answer will the question fetch?**

The question is a "How many?" question. For questions asking "how many", the answer should be a number.

**When is the data sufficient?**

The data is sufficient if we are able to get a UNIQUE answer for the number of positive numbers from the information in the statements.

If the statements do not have adequate data to uniquely determine how many among the three numbers are positive, the data is NOT sufficient.

**Key data from the question stem**

Each of the three numbers x, y, and z are less than 10.

Step 2 of solving this GMAT DS question:

Evaluate Statement (1) ALONE: x + y + z = 20

From the question stem we know that each number is less than 10.

So, x < 10, y < 10 and z < 10.

Therefore, the __maximum sum__ of any two of these numbers, say x + y < 20.

However, statement 1 states x + y + z = 20.

Unless the third number, z in this case, is also positive x + y + z cannot be 20.

Hence, we can conclude that all 3 numbers x, y and z are positive.

**Statement 1 ALONE is sufficient.**

Eliminate choices B, C and E. Choices narrow down to A or D.

Step 3 of solving this GMAT DS question:

Evaluate Statement (2) ALONE: x + y = 14

As each of x and y is less than 10, both x and y have to be positive for the sum to be 14.

However, z could also be positive or z could be negative.

So, there could be either 2 or 3 positive numbers among the three numbers.

We are not able to find a unique answer from the information in statement 2.

**Statement 2 ALONE is NOT sufficient.**

Eliminate choice D.

Statement 1 ALONE is sufficient. Choice A is the answer.

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