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kiran120680
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Joined: 18 Feb 2019
Last visit: 27 Jul 2022
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Location: India
Schools: AGSM '20 EDHEC Sept 2019 Intake Great Lakes '21 IE
GMAT 1: 460 Q42 V13
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The product of two positive integers is 96. The sum of the two number is 22. Which of the following could be the difference of the two integers?
A. 2
B. 5
C. 6
D. 8
E. 10
A. 2
B. 5
C. 6
D. 8
E. 10
\(ab = 96 = 2^5 * 3\)
\(a + b = 22\)
A few attempts will lead you to the right answer. In this case, \(96 = 16 * 6\) and \(16 + 6 = 22\). Hence, \(16 - 6 = 10\).
\(a + b = 22\)
A few attempts will lead you to the right answer. In this case, \(96 = 16 * 6\) and \(16 + 6 = 22\). Hence, \(16 - 6 = 10\).
Let the two integers be x and y.
Hence from the question we know that,
xy=96
x+y=22
Or y=22-x
Hence xy = 96
can be written as
x(22-x)=96
=>22x-x(sq)=96
=>x(sq)-22x+96=0
=>x(sq)-16x-6x+96=0
=>x(x-16)-6(x-16)=0
=>(x-16)(x-6)=0
=>x=16 or x=6
Hence, y=16 or y=6 respectively
Difference between the two integers[x-y or y-x] gives us 16-6=10
Hence, option E.
Hence from the question we know that,
xy=96
x+y=22
Or y=22-x
Hence xy = 96
can be written as
x(22-x)=96
=>22x-x(sq)=96
=>x(sq)-22x+96=0
=>x(sq)-16x-6x+96=0
=>x(x-16)-6(x-16)=0
=>(x-16)(x-6)=0
=>x=16 or x=6
Hence, y=16 or y=6 respectively
Difference between the two integers[x-y or y-x] gives us 16-6=10
Hence, option E.
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Expert reply
Posts: 22,737
kiran120680
We can create the equations in which x = one number and y = the other number:
xy = 96
and
x + y = 22
x = 22 - y
Substituting, we have:
(22 - y)y = 96
22y - y^2 = 96
y^2 - 22y + 96 = 0
(y - 16)(y - 6) = 0
So y = 16 or 6
When y = 16, x = 6, or when y = 6, x = 16.
So the difference could be 10.
Alternate Solution:
Letting x and y denote the numbers, we are given that x + y = 22 and xy = 96. Let’s find (x + y)^2:
(x + y)^2 = 484
x^2 + 2xy + y^2 = 484
Let’s subtract 4xy = 384 from each side of this equation:
x^2 + 2xy + y^2 - 4xy = 484 - 384
x^2 - 2xy + y^2 = 100
(x - y)^2 = 100
x - y = 10 or x - y = -10
Answer: E
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Products of 96
(1*96,2*48,3*32,4*24,6*16)
Only adding 6 and 16 gives 22.
Therefore the difference is 10
Posted from my mobile device
(1*96,2*48,3*32,4*24,6*16)
Only adding 6 and 16 gives 22.
Therefore the difference is 10
Posted from my mobile device
i solved it on the basis of logic
Stmt 1 - xy = 96 --Which means either x or y is even
Stmt 2 - x + y = 22 ---Since sum is even therefore both x and y even.
Now unit digit of 96 is 6. Therefore look for number that satisfy these conditions. You will arrive at 16,6 in a jiffy
Stmt 1 - xy = 96 --Which means either x or y is even
Stmt 2 - x + y = 22 ---Since sum is even therefore both x and y even.
Now unit digit of 96 is 6. Therefore look for number that satisfy these conditions. You will arrive at 16,6 in a jiffy
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