Can you prove that an odd number times an even number always gets an even number?

Question

e.g. 2*3 = 6 (even), 5*4 = 20 (even) ... however, is there an algebraic proof for this?

Answer 1

Let m and n be two arbitrary integers.
Then (2n + 1) is an odd number and 2m is an even number.
The product = (2n+1)*2m = 2{m(2n+1)} = 2k
where k = m(2n+1) is also an integer
So 2k is even.
Thus the product is always even.

Answer 2

Well this is true by definition.
An even number is a number with atleast one factor of 2, then for any general natural n, an even number may be written as , 2n
an odd number will be in the form (2k+1) for some general k. Note that 2 is NOT a factor here. Take the multiple,
2n . (2k+1) = 2[n.(k+1)]

2 will ALWAYS be a factor of the multiple, thereby the product yields an even number every time.

Hope this helps!!!!

Answer 3

It's probably easiest is you consider an even number as a number that if you divide it by two, the answer is an integer. Hence, if a number is even it can be written as 2n, where n is an integer and half of the original number. Multiplying 2n by ANY integer x, odd or even, would give you 2nx, or 2(nx). Since both n and x are integers, so is nx. This product is 2 times that integer nx, therefore it's even.

Answer 4

Yes, any even number can be considered as 2 X something. Let us make an odd number Y, and even number as 2B.

We can express the formula as following.

Y X 2B = 2 X Y X B = 2 (Y X B), therefore the answer is even.

I hope this answers your question. Thanks.

Answer 5

Sure. Any odd factor multiplied by an even factor and the product is always even.
Ex:
8*7=56
6*7=42

Answer 6

let,
odd number= 2m-1
even number= 2n

thus multiplication results
2n(2m-1)
=2z [z=n(m-1)]

surely any number multiplied by 2 is even, thus all debates end and an odd number times an even number always gets an even number. [proved]

Answer 7

any number with a factor of 2 is even. so when you multiply an even number which has factor 2, by an odd number, it will still have factor 2, therefore be even

algebraically (a,b positive integers)

even: 2a
odd: 2b-1

so, even*odd = 2a(2b-1) = 2K which is even, where K=a(2b-1)

Answer 8

Sure.... write the even number as 2x. Then, 2x times anything will have 2 as a multiple, therefore it will be even.

Answer 9

yes

even number= 2k
odd number = 2r+1
so teh product is:
2k(2r+1) = 2( 2rk+ k),
which is also a multiple of 2

Answer 10

4x3
2x5
2x9
8x7
7x6
6x5
8x9
9x10
7x10
5x10
3x10