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# The positive integers r, s, and t are such that r is

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Joined: 03 Mar 2010
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Schools: Simon '16 (M)
The positive integers r, s, and t are such that r is [#permalink]

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07 Sep 2011, 02:19
00:00

Difficulty:

65% (hard)

Question Stats:

40% (01:43) correct 60% (01:12) wrong based on 15 sessions

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The positive integers r, s, and t are such that r is divisible by s and s is divisible by t. Is r even?
(1) st is odd.
(2) rt is even.
[Reveal] Spoiler: OA

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Kudos [?]: 360 [0], given: 22

Math Forum Moderator
Joined: 20 Dec 2010
Posts: 1964

Kudos [?]: 2050 [0], given: 376

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07 Sep 2011, 02:40
The positive integers r, s, and t are such that r is divisible by s and s is divisible by t. Is r even?
(1) st is odd.
(2) rt is even.

r/s->Integer
s/t->Integer

(1) st is odd.
Both s and t are odd.

r=9; s=3; t=1
r=6; s=3; t=1
Not Sufficient.

(2) rt is even.
Either r or t or both are even.
r=even. Fantastic.
t=Even; s becomes even; t has to be even.

See this:
s/t=Integer; t=Even; s=Even*Integer=Even;
r/s=Integer; s=Even; r=Even*Integer=Even;

So, r is definitely even.
Sufficient.

Ans: "B"
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07 Sep 2011, 02:44
(1)

st is odd means that s and t are odd, but r can be even or odd

e.g r = 10, s = 5, t = 1

r = 15, s = 5, t = 1

Insufficient

(2)

If rt is even then at least one of t or r is even

So r = k*s

s = m*t

=> r = k*m*t (where k and m are positive integers)

=> rt = r * r/(km) = r^2/km is an even integer (as per question)

=> r^2 is even

=> r is even

Sufficient

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Kudos [?]: 592 [0], given: 36

Re: Even or Odd   [#permalink] 07 Sep 2011, 02:44
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