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

Question

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.

Bunuel · Accepted Answer

(1)  st=odd , clearly not sufficient as no info about  r , for example if  r=6 ,  s=1  and  t=1  then answer is YES but if  r=3 ,  s=1  and  t=1  then the answer is NO.

(2)  rt=even . For product of 2 integers to be even either one or both must be even. Can  r  not to be even? The only chance would be if  t  is even and  r  is odd. Let's check if this scenario is possible: if  t  is even, so must be  s , as  s  is divisible by  t  (if an integer is divisible by even it's even too). Now, if  s  is even so must be  r  by the very same reasoning. So scenario when  r  is not even is not possible -->  r=even . Sufficient.

Answer: B.

mehdiov · Answer

many thanks looks easy after the explanation

Do you have an idea about the level of this question ?

Bunuel · Answer

Not very hard (600+) but tricky, as it's C-trap question: the question which is obviously sufficient if we take statements together. When we see such questions we should become very suspicious.

frank1 · Answer

thanks...i was able to get to B but may be in 3 minutes.....
i complicated the question thinking like 2 4 8 and not thinking infact one can be one number or 2 numbers can be same 8 2 2 and so on...

fluke · Answer

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"

subhashghosh · Answer

(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

Answer - B

PathFinder007 · Answer

HI Bunnel,

I have a doubt on this.

Generally we treat both the statements as seprate statements. then why are you mixing them.

If I will go with st2 i can r can be even or odd because rt = even ( r and t both can be even or one of them is even) now if we refer even to r and t then st1 will contradict.

is this the reason you are not considering both r and t as even?

Please clarify

Thanks.

Bunuel · Answer

The statements do not contradict: st is odd and rt is even is possible when r is even and both s and t are odd.

skysailor · Answer

This is how I solved it:

Is r even?

r is divisible by s, hence r = ns
s is divisible by t, hence s = mt
combining, r = nmt

1-> st = odd. This implies, both s and t are odd. no impact on r. insuff.
2 -> rt = even
 Hence, r = even or t = even or both = even
 If t = even, then r = nmt = even
 If r = odd, then r is even for rt = even
 Suff.

harikrish · Answer

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.

Solution:

The Tricky part in this question is to remember that r,s and t are integers.

Statement 1: No information about r is given. r can be either even or odd. Insufficient.

Statement 2: r has to be even as Odd number divided by odd will never yield a even number and Odd number divided by even will not yield a integer.
Therefore r must be even.

Answer :Option B.

roadrunner · Answer

r = sx
s = ty

r = tyx

1. st is odd
O = O*y ==> y = odd
r = O*O*x.. we don't know x. Therefore, not sufficient

2. rt = even
r = tyx
O=E*y*x or
E = O*y*x

evaluate whether both outcomes are possible.
when r is odd
t, y, and x must be odd which is not possible since t is even in this case.

thus, r is even

Ans. B

Nunuboy1994 · Answer

The thing about these problems is that even though they appear to be very simple math they are designed to be very misleading

Statement 1

you could have 12 3 1 or 9 3 1

insuff

Statement 2

R cannot be odd because t has to be a multiple of 2- - because t has to be a factor of r

suff

B

zanaik89 · Answer

What happens when t is odd?

Bunuel · Answer

For (2) if t is odd then r must be even right away, because rt=even.

exc4libur · Answer

Rule:
EVEN/EVEN = anything {not defined (divisibility by 0), even, odd, proper fraction}
ODD/ODD = odd or proper fraction
EVEN/ODD = even or proper fraction
ODD/EVEN = not defined or proper fraction

Given: if r/s=integer and s/t=integer, then r/t=integer.

(1) st is odd: then, s and t are both odd, only ODD*ODD=ODD; from this, we have two cases, 
r/ODD=integer, then r=ODD/ODD, or r=EVEN/ODD, different answers insufficient.

(2) rt is even: then, r and t cannot be both odd; however, r=ODD/t=EVEN is not an integer,
so the only valid cases are r=EVEN/t=EVEN, and r=EVEN/t=ODD, sufficient.

Answer (B).

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