If m, p, and t are positive integers and m<p<t, is the

If m, p, and t are positive integers and m < p < t, is the product mpt an even integer?

For \(mpt\) to be even at least one should be even (as m, p, and t are integers).

(1) \(t-p=p-m\):

\(\frac{t+m}{2}=p\)

This algebraic expression means that \(p\) is halfway between \(t\) and \(m\) on the number line:

\(--m---p---t--\)

So m, p, and t are evenly spaced. Does this imply that any integer must be even? Not necessarily. If \(p\) is odd and \(m\) and \(t\) are some even constant below and above it, then all three will be odd. So we can have an YES as well as a NO answer. For example:

If \(m=1\), \(p=3\), \(t=5\) the answer is NO;
If \(m=2\), \(p=4\), \(t=6\) the answer is YES.

Not sufficient.

(2) \(t-m=16\). Clearly not sufficient. No info about \(p\).

(1)+(2) Second statement says that the distance between \(m\) and \(t\) is 16, so as from (1) \(m\), \(p\), and \(t\) are evenly spaced, then the distance between \(m\) and \(p\) and the distance between\(p\) and \(t\) must 8. But again we can have two different answers:

\(m=0\), \(p=8\), \(t=16\) --> \(mpt=even\);
\(m=1\), \(p=9\), \(t=17\) --> \(mpt=odd\).

Two different answers. Not sufficient.

Answer: E.

Hope it's clear.
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Ok this is my take, take statement 1, t+m/2= p which means that t+m/2 equals an integer and the only way that is possible is when t+m are either both odd or both even since if they are odd and even that doesn't given an integer. Secondly, if t+m is either both odd or both even the result will be even hence p= even. But statement 1 insufficient because we dont know whether t+m are both odd or even.

There BCE
Now let's take statement 2

We have no information about p so insufficient.
Hence CE
Now both statements taken together
We still can't conclude that t and m are either odd or even since that will determine whether the product is even. So E.
Do let me know if my solution makes sense to people.

Posted from GMAT ToolKit

Hey!

I need help with this one.

It's obvious that each statement alone is not sufficient, but I'm struggling with both statements together not sufficient.

From (1) we know that p=(t+m)/2
From (2) we know that t-m=16 -> following this info, you can say that either both t and m are even or both are odd.

Now putting it together, since t+m will be even and and saying that (t+m)/2 will be even, too, we can say that it will be even?!?!

Not sure understood your question correctly. Anyway, check the examples provided in this post: if-m-p-and-t-are-positive-integers-and-m-p-t-is-the-product-mpt-an-even-integer-126259.html#p729805 Hope they'll help to clear your doubts.
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We need to find whether at least one of m, p and t is even.

S1: p = (t+m)/2 tells us that (t+m) is even since p has to be integer. So all we know is that t and m are both either odd or both even (since their sum is even). It doesn't say anything about p i.e. whether p is even or odd. p could be odd e.g. (4+2)/2 = 3 or (5+1)/2 = 3 or it could be even e.g. (8+4)/2 = 6 or (7+5)/2 = 6 etc.

S2: t - m = 16 tells us that t and m are either both odd or both even (because the difference between them is even).We again don't know whether they are even. e.g. 18 - 2 = 16 or 17 - 1 = 16. We also don't know whether p is even.

Using both statements, 2p = t+m, 16 = t - m. Add them to get p+8 = t. If t is odd, p is also odd. If t is even, p is also even. So basically, all 3 variables are either odd or all three are even. But we do not know whether they are even. Hence not sufficient.
Answer (E)
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Bunuel, great explanation but in my humble opinion I don't think you can take m=0 since it says that m p and t are positive integer

Correct, but it does not affect the answer. Consider:
\(m=2\), \(p=10\), \(t=18\) --> \(mpt=even\);
\(m=1\), \(p=9\), \(t=17\) --> \(mpt=odd\).
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Hi,

Can someone help me find where I'm wrong in solving the following Q:

Q: If m,p and t are positive integers and m<p<t, is the product mpt an even integer?
1. t-p=p-m
2. t-m= 16

O.A (E)

My interpretation:

Q: For "mpt" to be an even integer atleast one of the three numbers should be even. to find if any one of m,p or t is even.

1. t-p=p-m

so, t+m=2p
t+m/2 = p

if t+m is divisible by 2 and results in an integer 'p', then 'p' has to be a multiple of '2' which is even. hence 'mpt' should be even. SUFFICIENT

we are down to Options A or D.

2. t-m=16.

this only says the diff of t and m is even. so t and m are either both 'odd' or both 'even'. INSUFFICIENT.

So the correct Answer ( in my opinion) is A.

kindly advise.

Hi. please post the OA inside the spoilers to give other a fair shot at the problem. Thanks.

you have said -> "if t+m is divisible by 2 and results in an integer 'p', then 'p' has to be a multiple of '2' which is even. hence 'mpt' should be even. SUFFICIENT"

P can or cannot be a multiple of 2, t+m is a multiple of 2.
Take t=3 m =3 then t+m = 6, which is multiple of 2, and p can be 3 so that t+m = 2 (p) = 3+3 = 2 * 3
tmp = odd
if we take t = m = 6 then tmp = Even
t+m = 2p then only 2 things we can deduct
both t and m are both odd or Even, as E+O would result in Odd,which cannot be a multiple of 2

HTH

Cheers

1) t-m = 2p
or t-m = even
so, t,m = (even,even) or (odd,odd)
and p can be even or odd
so A alone is not sufficient.

2) t-m = 16
t,m = (17,1) or (18,2) ....
again the same condition and p is unknown.
so B alone is also insufficient.

(1)+(2)
t+m = 2p
t-m = 16
solving; t-p = 8
t,p = (100,92) or (101,93) ...
when (t,p) = (100,92) then m = even and m*t*p = even
when (t,p) = (101,93) then m = odd and m*t*p = odd
hence (1)+(2) also is insufficient.

Therefore, answer is E.

------------------
+1 if you liked my solution. Tx.
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We are given that m, p, and t are positive integers with m < p < t, and we need to determine whether mpt is an even integer.

Statement One Alone:

t – p = p – m

Simplifying the equation in statement one, we have:

t – p = p – m

t + m = 2p

Since 2p must be even, we see that either t and m are both odd or t and m are both even. However, since we know nothing about p, p could be either odd or even. If t and m are both even, then regardless of whether p is odd or even, the product mpt will be even. However, if t and m are both odd and p is also odd, then mpt will be odd. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

t – m = 16

Since t – m = 16, we see that either t and m are both odd or t and m are both even. However, since we know nothing about p, statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using statements one and two, we still know only that either t and m are both odd or t and m are both even, and we still have no information regarding p. Thus, the two statements together are not sufficient.

Answer: E
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