If p, x, and y are positive integers, y is odd, and p = x^2 + y^2, is x divisible by 4?
(1) When p is divided by 8, the remainder is 5.
(2) x - y = 3
(1) When p is divided by 8, the remainder is 5.
(2) x - y = 3
An odd integer can be expressed in the following form:Mo2men wrote:If p, x, and y are positive integers, y is odd, and p = x^2 + y^2, is x divisible by 4?
(1) When p is divided by 8, the remainder is 5.
(2) x - y = 3
2k + 1.
Since y is odd, y = 2k + 1.
Statement 1:
In other words, p is 5 more than a multiple of 8:
p = 8a + 5.
Since p = x² + y² and p = 8a + 5, we get:
x² + y² = 8a + 5
x² + (2k+1)² = 8a + 5
x² + 4k² + 4k + 1 = 8a + 5
x² + 4k² + 4k = 8a + 4
x² = 8a + 4 - (4k²+4k)
x² = 4(2a+1) - 4k(k+1)
x² = 4[2a+1 - k(k+1)].
As shown above, 2a+1 = odd integer.
Since k(k+1) is the product of two consecutive integers, k(k+1) = even.
Thus, the resulting equation above implies the following:
x² = 4(odd - even)
x² = 4(odd)
x = 2(√odd)
x = 2(odd).
A multiple of 4 can be divided twice by 2.
Since x = 2(odd) CANNOT be divided twice by 2, x is NOT a multiple of 4.
SUFFICIENT.
Statement 2:
If x=4 and y=1, then x is a multiple of 4.
If x=5 and y=2, then x is NOT a multiple of 4.
INSUFFICIENT.
The correct answer is A.
