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16 Sep 2014, 01:44
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If \(0 \lt x \lt y\) and \(x\) and \(y\) are consecutive perfect squares, what is the remainder when \(y\) is divided by \(x\)?
(1) Both \(x\) and \(y\) have 3 positive factors.
(2) Both \(\sqrt{x}\) and \(\sqrt{y}\) are prime numbers.
(1) Both \(x\) and \(y\) have 3 positive factors.
(2) Both \(\sqrt{x}\) and \(\sqrt{y}\) are prime numbers.
16 Sep 2014, 01:44
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Official Solution:
If \(0 \lt x \lt y\) and \(x\) and \(y\) are consecutive perfect squares, what is the remainder when \(y\) is divided by \(x\)?
Observe that, since \(x\) and \(y\) are consecutive perfect squares, it follows that \(\sqrt{x}\) and \(\sqrt{y}\) are consecutive integers. For example:
4 and 9 are consecutive perfect squares, and their square roots, \(\sqrt{4}=2\) and \(\sqrt{9}=3\), are consecutive integers.
16 and 25 are consecutive perfect squares, and their square roots, \(\sqrt{16}=4\) and \(\sqrt{25}=5\), are consecutive integers.
(1) Both \(x\) and \(y\) have 3 positive factors.
Since only the squares of primes have three factors (for example, the factors of \(p^2\), where \(p\) is a prime, are 1, \(p\), and \(p^2\)), this statement implies that \(x=(prime_1)^2\) and \(y=(prime_2)^2\). Based on the previous observation that the square roots of consecutive perfect squares are consecutive integers, \(\sqrt{x}=prime_1\) and \(\sqrt{y}=prime_2\) must be consecutive integers. The only pair of consecutive integers that are both primes is 2 and 3. Therefore, \(x=(prime_1)^2=4\) and \(y=(prime_2)^2=9\). When dividing 9 by 4, the remainder is 1. Sufficient.
(2) Both \(\sqrt{x}\) and \(\sqrt{y}\) are prime numbers.
Using the same logic as above, since the only pair of consecutive integers that are both primes is 2 and 3, we can conclude that \(\sqrt{x}=2\) and \(\sqrt{y}=3\). Therefore, \(x=4\) and \(y=9\). When dividing 9 by 4, the remainder is 1. Sufficient.
Answer: D
If \(0 \lt x \lt y\) and \(x\) and \(y\) are consecutive perfect squares, what is the remainder when \(y\) is divided by \(x\)?
Observe that, since \(x\) and \(y\) are consecutive perfect squares, it follows that \(\sqrt{x}\) and \(\sqrt{y}\) are consecutive integers. For example:
4 and 9 are consecutive perfect squares, and their square roots, \(\sqrt{4}=2\) and \(\sqrt{9}=3\), are consecutive integers.
16 and 25 are consecutive perfect squares, and their square roots, \(\sqrt{16}=4\) and \(\sqrt{25}=5\), are consecutive integers.
(1) Both \(x\) and \(y\) have 3 positive factors.
Since only the squares of primes have three factors (for example, the factors of \(p^2\), where \(p\) is a prime, are 1, \(p\), and \(p^2\)), this statement implies that \(x=(prime_1)^2\) and \(y=(prime_2)^2\). Based on the previous observation that the square roots of consecutive perfect squares are consecutive integers, \(\sqrt{x}=prime_1\) and \(\sqrt{y}=prime_2\) must be consecutive integers. The only pair of consecutive integers that are both primes is 2 and 3. Therefore, \(x=(prime_1)^2=4\) and \(y=(prime_2)^2=9\). When dividing 9 by 4, the remainder is 1. Sufficient.
(2) Both \(\sqrt{x}\) and \(\sqrt{y}\) are prime numbers.
Using the same logic as above, since the only pair of consecutive integers that are both primes is 2 and 3, we can conclude that \(\sqrt{x}=2\) and \(\sqrt{y}=3\). Therefore, \(x=4\) and \(y=9\). When dividing 9 by 4, the remainder is 1. Sufficient.
Answer: D
General Discussion
19 Jun 2018, 02:05
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Bunuel
D was very prominent , but I overthought ,
1st statement : 3 positive factors , this means the number is prime . But in this case , 5 & 11 can also be a option ? even 5 & 7 ? If y is 7 , and x is 5 then the remainder is 2 , and if y is 3 and x is 2 , then the remainder is 1 . This is the only confusion I'm having .
other that both the statement clearly indicates that x and y are prime .
Statement 2 , clearly says , that x & y are prime , as both are perfect squares
I chose E , as I overlooked that X & Y are consecutive perfect square. I was determined that X & Y both are prime, Now when I'm reading it again , it bloody says consecutive perfect square , hence 2 & 3 are the only option. Makes perfect sense.
Nice question Bunuel .
