A. Statement 1 ALONE is sufficient, but statement 2 alone is
not sufficient to answer the question asked.

B. Statement 2 ALONE is sufficient, but statement 1 alone is
not sufficient to answer the question asked.

C. BOTH statements (1) and (2) TOGETHER are sufficient to
answer the question asked; but NEITHER statement ALONE is sufficient.

D. EACH statement ALONE is sufficient to answer the question
asked.

E. Statements (1) and (2) TOGETHER are NOT sufficient to answer
the question asked, and additional data specific to the problem are needed.

##
**Data Sufficiency Problems for GMAT – Number Theory Problem 16**

If a
and b are integers, is a+b-1 is a multiple of 3?

1)
When a is divided by 3, the remainder is 2

2)
When b is divided by 6, the remainder is 5

**Answer: C**

**Explanation:**

*Statement (1)***alone**given when a is divided by 3, it gives the remainder 2.

The value of a must be greater than a multiple of 3 by
exactly 2: The possible value for are 5,8,11,14,17,20,

It provides information about a. We don’t have
any information about b.

So statement I alone is not sufficient to
answer the question.

**Statement (2) alone->**Given that ,when b is divided by 6, it gives the remainder 5.Then the value of b must be greater than a multiple of 6 by exactly 5: The possible values of b are 11,17,23,29,…

It provides information about the b. It does
not provide any information about the other variable a.

So statement II alone is not sufficient to
answer the question

**Consider both statements (1) and (2)**together, adding any a-value and any b-value, always results in a sum that exceeds a multiple of 3 by exactly 1 or 7.

Subtracting 1 from that sum, always results in
a multiple of 3.

So both statements together are needed to
answer the question.

##
**GMAT Solved Problems on
Data Sufficiency – Number System Problem 17**

If p, q, r and s are positive integers, is the
sum of pq and rs an even integer?

(1) p and r are both even integers.

(2) q is an odd integer and s is an even integer.

**Answer :A**

**Explanation:**

**Statement 1 alone**: - Given that p and r are even integers, The product of an even integer and any other integer is always even. Therefore, pq and rs also even integers,

So we can say that
sum of pq and rs is even (The sum of two even
integers is always even)

Statement 1 alone is sufficient to answer the
question.

**Statement 2 alone,**Given q is odd and s is even.

s is even , therefore rs is even.

When q is odd, pq is even when p is even and
pq is odd when p is odd.

We don’t have nature of p in statement ii
alone.

So pq can be odd or even

When pq is odd, sum of pq and rs is odd

When pq is even, sum of pq and rs is even

So statement ii alone is not sufficient to
answer the question.

##
**Solved Problems on Data
Sufficiency – Number System Problem 18**

If x +
2y = 12, then what is the value of y?

(1) x =y-3

(2) x is an even prime number

**Answer : D**

**Explanation :**

**From**

**statement 1 alone,**To know the value of two unknown variables, we need two equations containing those variables.

In the problem we have, x+2y=11

In statement I we have
y-x=1

Solving two equations, we can answer the
question.

**From statement 2 alone,**x is an even prime number. We have unique value for x i.e, 2.

Substituting the value of x in the equation,
we can find the value of y.

So statement ii alone, sufficient to get the
answer.

##
**Data Sufficiency Problems
on Numbers – Aptitude Solved Problem 19**

Is $\sqrt{a}$
greater than b?

(1) a = 5

(2) b is the smallest
prime number.

**Answer C**

**Explanation:**

**Statement 1 alone**, we have only information about a. a=5 then $\sqrt{a}$ =2.23..

We don’t have
information about b, so we cannot answer the question

**Statement 2 alone**, the b is the smallest prime number.

b=2.

It alone gives only
information about b.

The smallest prime is 2, which is less than 2.23. Consequently, $ \sqrt{a}$
is greater than b.

##
**GMAT Data Sufficiency Problems on Numbers –
Aptitude Solved Problem 20**

Is N

^{2}an even integer?
(1) The factors of N are 3, 5.

(2) N has no even factor.

**Answer:B**

**Explanation :**

**The product of the factors 3 and 5 is 15. So N must be multiple of 15.The possible values of N are 15,30,45,60..**

If N is even then N

^{2}is even.
If N is odd, then N

^{2}is odd.
\Statement I alone
is not sufficient to answer the question.

**Statement 2 alone,**says N has no even factor. It is possible when N is odd number only.

\ We can say that N

^{2}is not an even integer.
Statement ii alone is sufficient to answer the
question.

##
**GMAT Data Sufficiency Problems on Numbers –
Aptitude Solved Problem 21**

Is the number a integer?

(1) a is not divisible by 2.

(2) a has 3 factors 3,5,and 11.

**Answer: B**

**Explanation:**

**From statement 1 alone ,**a is not divisible by 2.

The value a can be integer 3 or rational
number 3.5

Hence statement I alone is not sufficient

**Statement 2 alone.**

The product of the factors 3 x 5 x 11 is 605. a must be an integer that is an odd multiple of
105.

\
Statement ii alone is sufficient to get the answer.

##
**GMAT Data Sufficiency Questions
on Numbers – Aptitude Questions 22**

Given that a =b +c, where a, b and c are different positive integers, is a prime number?

(1). c = 5b

(2) b and c
are odd positive integers.

**Answer : D**

**Explanation :**

Given a=b + c

**From statement 1**, a=b+5b => a=6b.

We get 6 multiple as the value of a.

So a cannot be prime.

\
Statement I alone is sufficient to answer the question.

**From statement 2**,b and c are odd positive integers.

b+c will be even because sum of two odd integers is always even.

So the value of a will be even ( not 2 because b and c are
different odd integers , so sum two different odd positive is integers is
always greater than 2).

So statement ii alone sufficient to get the answer.

##
**Data Sufficiency Problem on
Number Theory – Data Sufficieny Example 23**

Is x a positive integer (x¹0)?

(1) x

^{2}= x^{3}^{}
(2) x = x

^{2}**Answer: A**

**Explanation:**

**From statement 1 ,**x

^{2}= x

^{3}

It happens only when the value of x is
positive (1)

So statement I is sufficient to answer
the question.

**Statement ii alone,**

**Whether x is positive or negative, x**

^{2 }must be positive. So we cannot find the sigh of x.

So statement 2 alone is not sufficient to answer the question.

##
**Data Sufficiency Solved
Problems on Number System – Question 24**

The positive integer N is a perfect square; what is
its value?

(1) 50 < N < 100

(2) N is an odd number.

**Answer : C**

**Explanation :**

**From statement 1**, The number lies between 50 and 100 . It is perfect square . The possible values are 64,81. We don’t have unique value

Statement I alone is not sufficient to
answer the question.

**From statement 2**,N is an odd number. Again we get so many values for N like , 9,25,49,81,..

So statement ii alone is not sufficient
to answer the question.

Combining both statements I and ii
, we get unique value 81

So both statements are needed to answer
to question.

##
**Data Sufficiency Problems for GMAT ,CAT -
Number System Solved Problem 25**

If a and b
are positive integers, what is the
value of N?

(1) N=2

^{(a-b) }(2) a =b+5**Answer : C**

**Explanation :**

**From statement I alone,**We don’t have the values of a and b , so we cannot get the value of (a-b).So we can’t find the value of N

**Statement ii alone,**a=b+5 =>a-b=5

We here it is not given the relationship
between N and (a-b).

So statement ii alone is not
sufficient to answer the question.

Using both statements, N=2

^{(a-b) }from statement i
And (a-b) = 5 from statement ii .So we
can find the value of N if we use both statements together.

\Statements I and ii
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