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06 Apr 2021, 01:30
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If a and b are integers, and m is an even integer, is ab/4 an integer?
(1) a+b is even.
(2) m/(ab) is an odd integer.
(1) a+b is even.
(2) m/(ab) is an odd integer.
07 Apr 2021, 01:15
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If a and b are integers, and m is an even integer, is ab/4 an integer?
(1) a+b is even. This implies that either both a and b are odd or both even. Now, if a=b=1 the answer is NO but if a=b=2 the answer is YES. Not sufficient.
(2) m/(ab) is an odd integer. Since given that m=even, then we have that \(\frac{even}{ab}=odd\) --> \(ab*odd=even\) --> \(ab=even\) --> at least one of the unknowns is even. Not sufficient.
(1)+(2) Since from (2) we have that at least one from a and b is even then from (1) it follows that both a and b must be even. The product of two even numbers is divisible by 4. Sufficient.
Answer: C.
General Discussion
06 Apr 2021, 09:52
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1. If a+b is even then either both a&b are even or both a&b are odd.
-> If this is the case then, when both are even, 2*2/4 or 2*6/4 are both integers. But when both are odd 3*9/4 or 5*7/4 are not integers
---> 1. Insufficient
2. If (m) is even and (m)/ab = odd then we know that a*b is even (for example 20/5 = 4 vs 20/4 = 5). If ab is even than we know that a&b can both be even or a or b can be even and other can be odd.
-> if this is the case 3*2/4 not an integer but 8*2/4 is an integer. Alone this is not sufficient
---> 2. Insuffient
But when you put them together you get one case where both a&b are even. If this is the case than it has to be an integer.
Answer - C
-> If this is the case then, when both are even, 2*2/4 or 2*6/4 are both integers. But when both are odd 3*9/4 or 5*7/4 are not integers
---> 1. Insufficient
2. If (m) is even and (m)/ab = odd then we know that a*b is even (for example 20/5 = 4 vs 20/4 = 5). If ab is even than we know that a&b can both be even or a or b can be even and other can be odd.
-> if this is the case 3*2/4 not an integer but 8*2/4 is an integer. Alone this is not sufficient
---> 2. Insuffient
But when you put them together you get one case where both a&b are even. If this is the case than it has to be an integer.
Answer - C
