Pkit wrote:
DS , Q. 76 page 313
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
My solution is:
(1) t+m=2p ->, \((t+m)/2=p\), p can not be odd since:
if t and m are even, then p is even , keep in mind that m < p < t, \(m<>t\)
if t and m are odd, p is even, keep in mind that m < p < t, \(m<>t\),[(5+11)/2=8, but (5+5)/2=5 odd,but m<>t]
if t is even and m is odd, then p can not be odd , since (even+odd)/2 must give us
integer, so P could not be odd, thus it is even.
Then, if:
mt= even*even=even
mt= odd*odd=odd
mt=even*odd=even
Then we know that P must be even, so either of results m*t when multiplied by even number P give us Even product, so the product m*p*t= even
Sufficient.
(2) not sufficinet.
I choose A, however the OG12th's answer choice is
Am I wrong?
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.
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