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

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Re: The sum of two positive integers is 21. What is the value of the large [#permalink]

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-1The 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.
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Re: The sum of two positive integers is 21. What is the value of the large [#permalink]
Bunuel wrote:
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.

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