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05 Nov 2021, 10:10
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x and y are positive integers, and \(x\) − \(y\) > 0. If \(x\) + \(y\) has precisely three positive divisors, and \(x\) − \(y\) has precisely two positive divisors, what is the smallest possible number of positive divisors that \(x^2\) − \(y^2\) could have?
A) 2
B) 3
C) 4
D) 6
E) 8
A) 2
B) 3
C) 4
D) 6
E) 8
05 Nov 2021, 10:35
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nislam
x and y are positive integers, and \(x\) − \(y\) > 0. If \(x\) + \(y\) has precisely three positive divisors, and \(x\) − \(y\) has precisely two positive divisors, what is the smallest possible number of positive divisors that \(x^2\) − \(y^2\) could have?
A) 2
B) 3
C) 4
D) 6
E) 8
A) 2
B) 3
C) 4
D) 6
E) 8
Factor of x-y are 1 and x-y
x+y has 3 factors, 2 of which will be 1 and x+1.
For smallest number of positive divisors, x-1 May be the 3rd factor of x+y.
So the product of x-y and x+y will have minimum 4 factors , 1, x-y, x+y, x^2-y^2
Take x=3,y=1,
x-y or 2 has 2 factors 1,2
x+y or 4 has 3 factors 1,2,4
x^2-y^2 or 8 will have 4 factors 1,2,4,8
Answer is 4 Option C
Posted from my mobile device
07 Dec 2023, 06:21
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Can someone please explain how to go about this question algebraically? In case, plugging in numbers takes time - want to know a better approach.
Also, Abhishekgmat87 - can you please explain how you arrived at the below conclusion? I understood x+y has one factor as 1 and second factor as x+y. How is the third factor x-1 or x+1? I'm not following.
"x+y has 3 factors, 2 of which will be 1 and x+1.
For smallest number of positive divisors, x-1 May be the 3rd factor of x+y."
Factor of x-y are 1 and x-y
x+y has 3 factors, 2 of which will be 1 and x+1.
For smallest number of positive divisors, x-1 May be the 3rd factor of x+y.
So the product of x-y and x+y will have minimum 4 factors , 1, x-y, x+y, x^2-y^2
Take x=3,y=1,
x-y or 2 has 2 factors 1,2
x+y or 4 has 3 factors 1,2,4
x^2-y^2 or 8 will have 4 factors 1,2,4,8
Answer is 4 Option C
Posted from my mobile device
Also, Abhishekgmat87 - can you please explain how you arrived at the below conclusion? I understood x+y has one factor as 1 and second factor as x+y. How is the third factor x-1 or x+1? I'm not following.
"x+y has 3 factors, 2 of which will be 1 and x+1.
For smallest number of positive divisors, x-1 May be the 3rd factor of x+y."
Abhishekgmat87
nislam
x and y are positive integers, and \(x\) − \(y\) > 0. If \(x\) + \(y\) has precisely three positive divisors, and \(x\) − \(y\) has precisely two positive divisors, what is the smallest possible number of positive divisors that \(x^2\) − \(y^2\) could have?
A) 2
B) 3
C) 4
D) 6
E) 8
A) 2
B) 3
C) 4
D) 6
E) 8
Factor of x-y are 1 and x-y
x+y has 3 factors, 2 of which will be 1 and x+1.
For smallest number of positive divisors, x-1 May be the 3rd factor of x+y.
So the product of x-y and x+y will have minimum 4 factors , 1, x-y, x+y, x^2-y^2
Take x=3,y=1,
x-y or 2 has 2 factors 1,2
x+y or 4 has 3 factors 1,2,4
x^2-y^2 or 8 will have 4 factors 1,2,4,8
Answer is 4 Option C
Posted from my mobile device
