The sum of two positive integers is 21. What is the value of the large

Question

The sum of two positive integers is 21. What is the value of the larger integer?

(1) The product of the two integers is 104.
(2) The larger integer is a prime number.

KSBGC · Answer

the sum of the 2 positive integers is 21

we can prepare a short list of the integers whose sum is 21

1+20
2+19
3+18
4+17
5+16
6+15
7+14
8+13
9+12
10+11

these are the probable combinations.

statement 1: product of 2 integers is 104. only 8*13=104. so the larger one is 13. Sufficient. 
statement 2: the large integer is a prime number. we have couple of prime. it is impossible to specify. thus statement 2 is not sufficient.

NiharikaR · Answer

IMO A
ST 1
x+y=21
xy=104
x(21-x)=104
8,13

ST2
10,11 and 13,8

EgmatQuantExpert · Answer

Solution

Given:

Let us assume that the two positive integers are ‘x’ and ‘y’. 
 • x + y = 21-----------------(1)

To find:

• We need to find the value of the larger integer between x and y.

Statement-1  “ The product of the two integers is 104 “.

• x * y= 104
• By squaring on both the sides of equation (1), we get:
 o  (x + y) ^2 = 21^2 
o  x^2+y^2 + 2*x*y = 441 
o After subtracting 4xy on both the sides, we get:
   x^2+y^2 - 2*x*y = 441- 4(x * y)  
o  (x-y) ^2  = 441- 4* 104=25
o x-y =5 OR x-y= -5

Thus, we have two cases:

Case-1 )  x-y =5 and x + y =21

Adding both the equation, we get:
 • 2x= 26, x=13
• y= 8

The larger integer is x and its value is 13.
 
 Case-2 )  x-y = -5 and x + y =21

Adding both the equation, we get:
 • 2x= 16, x=8
• y= 13

The larger integer is y and its value is 13.

Since the value of the larger integer is same for both the cases,  Statement 1 alone is sufficient to answer the question.

Statement-2 : “ The larger integer is a prime number  “.

The value of (x + y) can be 21 for different values of x and y such that larger integer is a prime number.
 • For, x=11 and y=10, the value of x + y=21
• For, x=13 and y=8, the value of x + y=21
• For, x=17 and y=4, the value of x + y=21
• For, x=19 and y=2, the value of x + y=21

The value of the larger integer is different for different values of x and y. 
Hence,  Statement 2 alone is not sufficient to answer the question .

Hence, the correct answer is option A.
  Answer: A

GMATinsight · Answer

a+b = 21 
and a < b

Question:  b = ?

Statement 1 : The product of the two integers is 104

104 = 1*104 or 2*52 or 4*26 or  8*13

Sum is 21 is case 8 and 13 hence b = 13 hence

SUFFICIENT

Statement 2 : The larger integer is a prime number 
21 = 4+17 or 8+13 hence b may be 17 or 13 hence

NOT SUFFICIENT

Answer: option A

The sum of two positive integers is 21. What is the value of the large

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GMAT Data Sufficiency (DS) Questions

The sum of two positive integers is 21. What is the value of the large

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