Odds and Evens
If a is a negative odd number and b is a positive even number, then which of the following must be a positive even number?
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8 solutions
but the point is solving it with modular arithmetic
What is this logic for not even number(A negative times a positive is a negative, so ab won't work.)
odd negative integer can be written in the form of a= -(2x+1) and even positive integer can be written as b =2x. In case o -ab, it can be written as -(-(2x+1))(2x). After solving we get 2(2x*x+x). this is even as it is divisible by 2 and the solution is obvious to be even as ther isn't any negative sign. try it with other no.s. you wont get the answer.
therefore -ab is the answer
No need to complicate the solution like that! According to what we know, multiplication of an even and odd number always yields an even number, so we can say that a b is even. But, since signs of a and b are opposite, so a b is negative. To make it positive, we should multiply ( − 1 ) to the number. Thus, ( − a b ) is the answer.
If you want to verify, you can rule out the other options. Since, addition or subtraction between an even and an odd number always give odd number, so ( a + b ) and ( b − a ) can be ruled out. Now, a b , as we saw earlier is negative, not positive. So, ( − a b ) is the only correct answer.
ab will also be even
The question should be re-framed as:
Which of the following must be positive even number?
yes lol. same thing with me. i didnt know which one to choose
I have edited the question, sorry for my mistake.
Yes. I decided to pick the first one.
Let 2 k means a even number and 2 k + 1 means a odd number
1. − ( 2 k + 1 ) + 2 k = − 2 k − 1 + 2 k = 1 ,which is positive but odd
2. − ( 2 k + 1 ) ( 2 k ) = − ( 4 k 2 + 2 k ) = − 4 k 2 − 2 k = − 2 ( 2 k 2 + k ) = − 2 ( k ( 2 k + 1 ) ) ,which is even but negative
3. 2 k − − ( 2 k + 1 ) = 2 k + 2 k + 1 = 4 k + 1 = 2 ( 2 k ) + 1 ,which is positive but odd
4. − − ( 2 k + 1 ) ( 2 k ) = 4 k 2 + 2 k = 2 ( 2 k 2 + k ) = 2 ( k ( 2 k + 1 ) ) ,which is both
a=-(2n+1),b=2M so -ab=2M(2n+1)=2k K>O
a negative times positive is negative, so -negative(positive) * positive gets positive. odd times even is even, so -negative odd times positive even is positive even
let us assume that a value as -3 and b value as +2 which satisfies -ab=-(-3)*2=+6 which is a positive number
a= -1 b=2 therefore -ab=-(-1)(2)=2...2 is positive even number :)
An even number plus/minus an odd number is odd, so a + b and b − a won't work.
A negative times a positive is a negative, so a b won't work.
On the other hand, − a b is positive, and since b is even, the product will have 2 as a factor.