number theory is built into DS
for x^2 to be right square of integer, we need 350y to be right square too. Hence y will be some number 350 is missing to be square
prime factorization of 350,
350 (2) 175
175 (5) 35
35 (5) 7
7 (7) 1 ==> 2*(5^2)*7, we need one 2 and one 7 at least ... next correct square would be (2^2)*(5^2)*(7^2) or (2*5*7)^2 multiplied by some other squared number
y with 350 can and cannot be divisible by 28, we deal only with
y from 350y
let's turn to statements
st(1) x is divisible by 4, means that 350y square rooted would need to contain (2^2), hence our sqroot(350y) will be 2*5*7 multiplied by some other number containing 2, so that x and sqroot(350y) are divisible by 4. We answer Yes y is divisible by 28 and Sufficient.
st(2) x^2 is divisible by 28 means that y can and cannot be divisible by 28, depending on y's value. As y=2*7 it's not divisible by 28. But as y=2*7*(2^2)*(5^2)*(7^2) it's divisible by 28. Not Sufficient.
a
knight247 wrote:For positive integers x and y, x^2 = 350y. Is y divisible by 28?
(1) x is divisible by 4
(2) x^2 is divisible by 28
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