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If x and y are positive integers, is x^16 - y^8 + 345y^2 div

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If x and y are positive integers, is x^16 - y^8 + 345y^2 div [#permalink]

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If x and y are positive integers, is x^16 - y^8 + 345y^2 divisible by 15?

(1) x is a multiple of 25, and y is a multiple of 20
(2) y = x^2
[Reveal] Spoiler: OA
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Re: The relationship between factors and addition/subtraction? [#permalink]

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New post 29 Apr 2010, 08:58
thanatoz wrote:
So I know the basic rule is that, if a is a factor of both b and c, then a is a factor of (b + c)

But anything else?

Specifically, for a question like this on the math challenge. Is there a clear rule that will show quickly that since x and y are not multiples of 3, their addition cannot have it either? Thanks.

If \(x\) and \(y\) are positive integers, is \(x^{16} - y^8 + 345y^2\) divisible by 15?

1. \(x\) is a multiple of 25, and \(y\) is a multiple of 20
2. \(y = x^2\)

a is a factor of both b and c then a is a factor of b-c, b*c...
if x and y are not multiples of 3, when x^2n and y^2n are divided by 3, the remainder is 1.
1. That x is a multiple of 25 and y is a multiple of 20 doesn't give any clue of remainder of x,y when divided by 3 so insuf
2. suf
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Re: The relationship between factors and addition/subtraction? [#permalink]

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New post 04 May 2010, 14:32
Expert's post
thanatoz wrote:
So I know the basic rule is that, if a is a factor of both b and c, then a is a factor of (b + c)

But anything else?

Specifically, for a question like this on the math challenge. Is there a clear rule that will show quickly that since x and y are not multiples of 3, their addition cannot have it either? Thanks.

If \(x\) and \(y\) are positive integers, is \(x^{16} - y^8 + 345y^2\) divisible by 15?

1. \(x\) is a multiple of 25, and \(y\) is a multiple of 20
2. \(y = x^2\)


First of all as \(345y^2\) is divisible by 15 (this term won't affect the remainder), we can drop it.

The question becomes: is \(x^{16}-y^8\) divisible by 15?

(1) \(x\) is a multiple of 25, and \(y\) is a multiple of 20 --> they both could be multiples of 15 as well (eg x=25*15 and y=20*15) and in this case \(x^{16}-y^8=15*(...)\) will be divisible by 15 OR one could be multiple of 15 and another not (eg x=25*15 and y=20) and in this case \(x^{16}-y^8\) won't be divisible by 15 (as we can not factor out 15 from \(x^{16}-y^8\)). Not sufficient.

(2) \(y = x^2\) --> \(x^{16}-y^8=x^{16}-(x^2)^8=x^{16}-x^{16}=0\). 0 is divisible by 15. Sufficient.

Answer: B.

thanatoz wrote:
So I know the basic rule is that, if a is a factor of both b and c, then a is a factor of (b + c)

But anything else?

Specifically, for a question like this on the math challenge. Is there a clear rule that will show quickly that since x and y are not multiples of 3, their addition cannot have it either? Thanks.


If x and y both are not multiples of 3, then their sum or difference may or may not be multiple of 3:

x=2 and y=1 --> x+y=3 --> sum is multiple of 3;
x=2 and y=2 --> x-y=0 --> difference is multiple of 3;
x=2 and y=3 --> x+y=5 --> sum is not multiple of 3;
x=2 and y=0 --> x-y=2 --> difference is not multiple of 3.

BUT, if x is multiple of 3 and y is not (or vise-versa), then their sum or difference won't be multiple of 3.

Hope it helps.
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Re: If x and y are positive integers, is x^16 - y^8 + 345y^2 div [#permalink]

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New post 15 Aug 2013, 22:39
REM(x^16-y^8 +345*y^2)/15

REM(345/15)=0

So REM(x^16-y^8/15)

(1).

It may have different remainders for different values of x/y.
Insufficient

(2).

y=x^2
=> REM(x^16-x^16)/15
=>REM=0

=>Sufficient

(B) it is !
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Re: If x and y are positive integers, is x^16 - y^8 + 345y^2 div [#permalink]

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New post 26 Dec 2017, 22:26
From expression 345y^2 is certainly divisible 15 as 345/15 gives remainder 0.
we need to find if x^16 - y^8 is divisible by 15 or not.

From Statement 1:
If x=25 and y=20, then 25^16 - 20^8 = 5^8(5^24 - 4^8)/15 gives the remainder 10 ---> Not divisible
If x=75 and y=60, then 75^16 - 60^8 = 15^8(5^16 * 15^8 - 4^8)/15 gives the remainder 0 ---> divisible
Inconsistent result, so ST1 alone not sufficient.

From Statement 2:
y = x^2
(x^16 - x^16 + 345y^2) / 15 gives remainder 0 ---> Divisible.
so ST2 alone is sufficient.

Hence correct answer is B
Re: If x and y are positive integers, is x^16 - y^8 + 345y^2 div   [#permalink] 26 Dec 2017, 22:26
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