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X and y are integers, are they even?

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X and y are integers, are they even? [#permalink] New post 05 Oct 2006, 17:07
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PS: not sure of the answers

1)X and y are integers, are they even?

1). 2y-x=x^2-y^2

2). X is even.



IMO: C


2)M=O+P+Q, where O, P, and Q are consecutive positive integer; M=R*S*T, where R, S and T are positive consecutive integers. What is the remainder when M is divided by 5?
1). When O is divided by 5, the remainder is 1
2). When R is divided by 5, the remainder is 1

??
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 [#permalink] New post 05 Oct 2006, 18:48
For first question

X(X+1) = Y(Y+2)

LHS is always even since either X is even or X + 1 is even (X is odd in that case)

Thus RHS has to be even and Y has to be even

Thus statement 1 tells only about Y and two about X thus together the answer is C

Let me work on the second question
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 [#permalink] New post 05 Oct 2006, 19:20
For second question, the answer is D

P has to be of the form 5n + 1 so does r

the sum is 15 N +6 leaves a remainder 1 when divided by 5

and product = 125 n^3 + 150 n^2 + 60 N +6 leaves a remainder 1 when divided by 5
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 [#permalink] New post 05 Oct 2006, 19:42
O and R can have either 1 or 6 in unit's digit.
1)
if O is of the form X1 (i.e unit's digit 6, rest can be any numbers)
M= O+P+Q = X1+X2+X3 = Y6.
The unit's digit of M will be 6, so when M is divided by 5 remainder is 1.
if O is of the form X6:
M=O+P+Q = X6+X7+X8= Y1
remainder will be 1.
so, (1) is enough to answer

2) When R is of form X1
M=RxSxT = X1.X2.X3 = Z6 // so M%5 = 1
When R is of form X6:
M= X6.X7.X8 = Z6 // so M%5 = 1

hence D is the answer. Either of them is enough to answer
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 [#permalink] New post 06 Oct 2006, 11:28
1)X and y are integers, are they even?

1). 2y-x=x^2-y^2

2). X is even.

from one

2y-x (could be even or odd)

x^2 - y^2 ( could be even or odd).........insuff

from two

insuff

both together
x is even

thus 2y-x is even

and thus x^2 - y^2 = even - ??? = even thus y^2 is even and hus x,y even

my answer is C


2)M=O+P+Q, where O, P, and Q are consecutive positive integer; M=R*S*T, where R, S and T are positive consecutive integers. What is the remainder when M is divided by 5?
1). When O is divided by 5, the remainder is 1
2). When R is divided by 5, the remainder is 1

M=O+P+Q

FROM ONE IF O = 5X+1 THUS
P AND Q ARE 5X+2 , 5X+3

ADD THE REMAINDERS = 6 DEVIDE BY FIVE REMAINDER IS 1....SUFF

M=R*S*T

R = 5X+1 THUS S,T ARE 5X+2, 5X+3

(5X+1)(5X+2) = 25X^2+ 15X + 2

(25X^2+ 15X + 2)(5X+3) = 125X^3+75X^2+10X+75X^2+45X+6

THUS REMAINDERR IS 6/5 = 1 .......SUFF

MY ANSWER IS D
  [#permalink] 06 Oct 2006, 11:28
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