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## 01. Data Sufficiency Questions with Answers - NUMBER THEORY

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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.
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
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.
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.
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
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
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 y=0     . The number can 90 or 80or 70 ..
Using both statements, x-y=8
And xy=0
Ã°  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.
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
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.
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.
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, 1/3 x – 1/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
Explanation : Let x be the number
From statement I , (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
Explanation: Lets one’s and ten’s digit of the 2-digit number x and y respectively
From statement I, x2+y2=45 ---------(1)
From statement ii , y=x-3 ----------(2)
From statement iii,  (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
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
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.
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 n17 ?
I)             n is an even number
ii)            n is a prime number