# 01. Data Sufficiency Questions with Answers - NUMBER THEORY

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 on Number Theory Problem 1**

Is n an even number?

i) N is not multiple of 2

ii) N multiplied by an odd number gives an odd number

**Answer : D**

**Explanation :**From statement i) N is not multiple of 2 i.e., N is odd

Hence, the
statement I alone sufficient to answer the question

From statement ii,
N multiplied by an odd number gives an odd number

i.e., N x odd
number = Odd number

\N
is odd

Hence, statement ii
alone sufficient to answer the question

So statement I or
ii alone sufficient to answer the question.

**Data Sufficiency Problems on Number Theory Problem 2**

A, B and C are three consecutive even numbers (not necessarily
in same order). What is the sum of these numbers?

i)One-fourth of number is 24

ii) The difference between B and C is 4.

**Answer: C**

**Explanation :**

Given that , A , B
and C are three consecutive odd numbers(not necessarily in the same order)

From statement i)

1
4

of A=24 =>A=96
But don’t know the
position of A in three numbers

So statement I
alone not sufficient to get the answer

From statement ii)A
and C are first and third numbers(or
third and first numbers) respectively as their difference is 4..

But we don’t know
the values of A and C.

Combining the
statement I and ii, A=96 and the value of C can be 96-2 or 96+2.Hence we can
find the sum of A ,B and C by using both the statements together.

**Data Sufficiency Problems on Number Theory Problem 3**

What
is the number?

i) 20%
of that number is

1
5

th of that number
ii)
Three-fifth of that number is less by 18 of that number.

**Answer: b**

**Explanation:**

Let the number be
N.

From statement I ) 20% of N=

1
5

of N
We cannot get the
value of N

So statement I
alone not sufficient to answer the question

From statement ii)

3
4

N -18 =N
From this equation
we can find the answer

So statement ii
alone sufficient to get the answer

**Data Sufficiency Problems on Number Theory Problem 4**

What
is the 3-digit number?

i) The first and last digit of the numbers are 3,9

ii) The number is multiple of 9

**Answer: C**

**Explanation:**

From statement i) The
number is 3x9.

Statement I alone
not sufficient

From statement ii) The
number is multiple of 9 , means the sum
of the digits of the number is also multiple of 9

(Divisibility rule
of 9)

But statement 2
alone not sufficient

Combining both I
and ii , 3+ x+9 must be divisible by 9.

12+x must be divisible by 9.

So we can get the
answer using both statements.

**Data Sufficiency Problems on Number Theory Problem 5**

What
is the two-digit number?

i) The difference between the 2 digits of the number is 8

ii) The product of the 2 digits of the number is 0

**Answer : C**

**Explanation :**

Let greater and
smaller digits of the number be x and y respectively. Then

From statement I ,
x –y =8

The number can be
91 or 80. So we cannot answer question using statement 1 alone.

From statement ii,
x x y=0 . The number can 90 or 80 or 70
..

Using both statements,
x-y=8

And xy=0

ð X
=8 and y =0

So both statements
are required to answer the question

**GMAT Data Sufficiency Problems on Number Theory Problem 6**

What
is a 2 digit number?

i)The sum of the two digits of the number is 16.

ii)The difference between
the two digits of the number is 1.

**Answer : C**

**Explanation :**

From statement 1)
Sum of the digits is 16.

SO the can number
can be 97,88

So statement 1
alone is not sufficient

From statement 2,
the difference between the two digits of the number is 1

The number can be
98,87,76,65,…

Combining both
stements,

Sum of the digits
is 16.

We get 97,88

The difference
between the digits is 0

9-7 is 2

8-8 is equal to 0. The two digit number is 88

So both statements
are required to answer the question.

**GMAT Data Sufficiency Problems on Number Theory Problem 7**

What
is 2-digit number?

i) The sum
of the digits of the number is 9

ii)
The digit in the units place of half of the digit in the tens place

**Answer : C**

**Explanation:**

Let the digit at
units place is y and tens place is x

From statement I , x + y =9

We have so many
such two digit number like 81,72,63,54,..

So statement I
alone can’t answer the question

From statement ii,
y=

1
2

x
So we can’t answer using statement 2
alone

Combining both I
and ii statements,

From statement I,
x+y=9

From statement ii,
x=2y

=>3y=9

=>y=3 then x=2y =>x=6

So both statements
are required to get the answer

**Data Sufficiency Problems on Numbers Problem 8**

What
is the value of 2-digit number?

i)
The difference between the two digits is 1
and product of the digits is 56

ii)The
digit at the tens place is greater than the digit at the units place.

**Answer: C**

**Explanation:**

Let digits of the
2-digit number be x and y.

From
statement i: xy=56

x-y=1.

=>Thus,
the number can be either 78 or 87

So
statement alone cannot give the answer

From
statement ii, x>y

Statement
ii alone is not sufficient, because we don’t the values of x and y.

Using
both I and ii statements, we can find the number 87

So
we can answer the question using both statements.

**Data Sufficiency Problems on Number Theory Problem 09**

What
is the value of the number?

i)80%
of that number is four-fifth of that
number

ii)The
difference between the one third and one fourth of that number is 15.

**Answer : b**

**Explanation :**Let x be the number

*From statement I*, (

80
100

)x=(
4
5

)x
We
cannot answer the question using I alone

*From statement ii,*1x – 14 x=15

We
can find answer using the statement ii alone

**Data Sufficiency Problems on Number Theory Problem 10**

What
is value of 3-digit number?

i)Two
– third of that number is less by 50 of that number

ii)The
sum of the digits is 6

**Answer : a**

**Explanation**: Let x be the number

From statement I ,
($\frac{2}{3}$
)x =x - 50

We can find the
value of the number using statement I alone sufficient to get the answer

From statement ii,
Sum of the digits is 6.

We can have so many
such numbers

So statement ii
alone not sufficient to answer the question.

**Data Sufficiency Problems on Numbers Problem 11**

What is
value of 2 digit number ?

i)The
sum of the squares of the two digits is 45

ii)The
digit in the tens place is 3 less than the digit in units place

iii)The
ratio between the 2-digit number and the sum of the digit of that number is 4:1

a)
Only I and
ii together are sufficient

b)
Only I and
iii together are sufficient

c)
Any two of
the three together are sufficient

d)
Statement
iii alone sufficient

e)
None of the
above

**Answer : d**

**Explanation:**Lets one’s and ten’s digit of the 2-digit number x and y respectively

From statement I, x

^{2}+y^{2}=45 ---------(1)
From statement ii , y=x-3 ----------(2)

From statement iii, $\frac{10y+x}{x+y}$
=4:1 -----(3)

From any two of the above three equations, we can find
the values of x and y and hence we can find the two digit number.

**GMAT Data Sufficiency Problems on Number Theory Problem 12**

What
is the value of 2-digit number?

i) Difference between the two digits is 1

ii) Number obtained by interchanging the digit is less than the
original number by 9.

iii) Sum of the digits is 9.

a)
Only I and
iii

b)
Only I and
ii

c)
Only ii and iii

d)
All
statements together

e)
Data
inadequate

**Answer : C**

**Explanation :**

Lets one’s and
ten’s digit of the 2-digit number x and y respectively.

From statement
I, x –y =1

From statement
ii, (10y+x) –(10x+y)=8.

(9x-9y)=9

(x-y) =1

From statement iii,
x + y =9

Using statements I
and iii , we can get the values of x and y but it cannot determined which of
the two digits x and y is greater and which is smaller

Using statement ii
and iii, we can find the values of x and y ,and in statement ii it is clearly
stated that original number is larger than number obtained the interchanging
the digits.

Hence , only by
using ii and iii, we can get the number.

**Aptitude Data Sufficiency Problems on Number Theory Problem 13**

What
is the two –digit number?

i) The
digit in tens place is 5 more than digit in the units place

ii) The
difference between the two digits is 5

iii) The
difference between the number obtained by interchanging the digits and the
original number is is 45.

a)
Only I and
iii

b)
Only I and
ii

c)
Only ii and iii

d)
All
statements together

e)
None of
above

**Answer: e**

**Explanation :**

Lets one’s and
ten’s digit of the 2-digit number x and y respectively.

From statement I, y=x+5 =>
y-x =5

From statement ii ,
y-x=5

From statement iii
, (10y+x)-(10x+y)=45

y- x=5

If we see each of
the statements separately, we find that we get only the difference between the
two digits of the number.

Hence, we can find
the answer even by using all the three statements together.

**GMAT Data Sufficiency Problems on Number System Problem 14**

Five
integers A, B,C, D and E are arranged in descending order in such a way that there are two
integers between B and C and B is not the greatest. There exists one integer
between D and E.A is not the smallest integer. Which one is the third smallest? (MHT-CET MBA 2012)

i)
E is the
greatest

ii. There exists no integer between B and E.

**Answer: a**

**Explanation:**Using the given information in the question, We can arrange the integers in two ways

E B D A C or D B E
A C

From statement I
, the greatest integer is E. It is clear
that correct sequence is E B D A C. Thus D is third smallest.

**GMAT Data Sufficiency Questions on Number System Problem 15**

What
is value of n

^{17 }?
I)
n is an even
number

ii)
n is a prime
number

**Answer: c**

**Explanation:**From statement I, n Is an even number. Here we cannot find the unique value of n . n can be 2,4,6,8…

From statement ii,
n is a prime number. Again n cannot have
unique value . n can be 2,3,5 ,…

Combining I and ii,
n is even and prime number. The only even prime number is 2.

Hence both
statements I and ii are sufficient to answer the question

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