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02. GMAT Data Sufficiency Problems on Number Theory 2

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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.
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.
Considering both statements I and ii .
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 N2 an even integer?
(1) The factors of N are 3, 5.
(2) N has no even factor.
Answer:B
Explanation :
Statement 1 alone
 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 N2 is even.
If N is odd, then N2 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 N2 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 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) x2 = x3
(2) x = x2
Answer: A
Explanation:
From statement 1 ,  x2 = x3
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, x2  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 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 together needed to answer the question
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