小女数学不好，求高人解答数学题！

margaretfang23

If a and b are positiveintegers such that a – b and a/b are both evenintegers, which of the following must be an odd integer?
A a/2
B  b/2
C   (a+b)/2
D    (a+2)/2
E  (b+2)/2
求解答过程？如果遇到类似的题怎么思考？

jeffxu

My understanding,
（1）a-b = even => a=even and b=even; or a=odd and b=odd
（2）a/b = even

（1）+（2）=> a=even, b=even  What's more, a/b = even => a must be multiple of 4.
So (a+2)/2 = a/2 +1, a/2 must be even; so a/2+1 must be odd.

Key: D

ctuwuzida

My understanding,
（1）a-b = even => a=even and b=even; or a=odd and b=odd
（2）a/b = even

（1）+（2）=> a=even, b=even  What's more, a/b = even => a must be multiple of 4.
So (a+2)/2 = a/2 +1, a/2 must be even; so a/2+1 must be odd.

Key: D

-- by 会员 jeffxu (2011/8/15 19:51:09)

Why a must be multiple of 4?

jeffxu

My understanding,
（1）a-b = even => a=even and b=even; or a=odd and b=odd
（2）a/b = even

（1）+（2）=> a=even, b=even  What's more, a/b = even => a must be multiple of 4.
So (a+2)/2 = a/2 +1, a/2 must be even; so a/2+1 must be odd.

Key: D

-- by 会员 jeffxu (2011/8/15 19:51:09)

Why a must be multiple of 4?

-- by 会员 ctuwuzida (2011/8/15 19:58:00)

Do you agree that both a and b are even?
If so, because a/b is even =>a/b=2k (k is an integer) => a = 2k*b,  since b is even,say b=2n => a = 4kn

ctuwuzida

My understanding,
（1）a-b = even => a=even and b=even; or a=odd and b=odd
（2）a/b = even

（1）+（2）=> a=even, b=even  What's more, a/b = even => a must be multiple of 4.
So (a+2)/2 = a/2 +1, a/2 must be even; so a/2+1 must be odd.

Key: D

-- by 会员 jeffxu (2011/8/15 19:51:09)

Why a must be multiple of 4?

-- by 会员 ctuwuzida (2011/8/15 19:58:00)

Do you agree that both a and b are even?
If so, because a/b is even =>a/b=2k (k is an integer) => a = 2k*b,  since b is even,say b=2n => a = 4kn

-- by 会员 jeffxu (2011/8/15 20:09:38)

yeah, got it.thanks! it`s very tricky

margaretfang23

My understanding,
（1）a-b = even => a=even and b=even; or a=odd and b=odd
（2）a/b = even

（1）+（2）=> a=even, b=even  What's more, a/b = even => a must be multiple of 4.
So (a+2)/2 = a/2 +1, a/2 must be even; so a/2+1 must be odd.

Key: D

-- by 会员 jeffxu (2011/8/15 19:51:09)

Thanks!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

wyw1018

a-b is even integer ->a-b=2m,m=1,2,3...
a/b is even integer->a/b=2n->a=2nb, n=1,2,3...
plug a=2nb into a-b=2k we get b(2n-1)=2m->b must be even integer denotes b=2k,k=1,2,3...
plug b=2k into a=2nb ->a=4kn, k,n=1,2,3...
(1) a/2=2kn, must be even integer
(2) b/2=k, k=1,2,3... can be odd or even integer
(3) (a+b)/2=(4kn+2k)/2=k(2n+1), k,n=1,2,3... can be odd or even integer
(4) (a+2)/2=(4kn+2)/2=2kn+1, k,n=1,2,3... must be odd integer
(5) (b+2)/2=(2k+2)/2=k+1, k=1,2,3... can be odd or even integer

therefore, the answer is D.

wyw1018

遇到奇偶的问题，一律写成2n 或者2n+1 轻松解决问题

margaretfang23

遇到奇偶的问题，一律写成2n 或者2n+1 轻松解决问题

-- by 会员 wyw1018 (2011/8/17 5:30:31)

非常到位！

margaretfang23

My understanding,
（1）a-b = even => a=even and b=even; or a=odd and b=odd
（2）a/b = even

（1）+（2）=> a=even, b=even  What's more, a/b = even => a must be multiple of 4.
So (a+2)/2 = a/2 +1, a/2 must be even; so a/2+1 must be odd.

Key: D

-- by 会员 jeffxu (2011/8/15 19:51:09)

另外还想请教一道数学题：Q21:
If x and y are positive integers, what isthe value of (x+y)²?
(1)   x = y - 3
(2) x and y areprime numbers.
                 
A. Statement (1) ALONE is sufficient,but statement (2) alone is not sufficient.
B. Statement (2) ALONE issufficient, but statement (1) alone is not sufficient.
C. BOTH statements TOGETHERare sufficient, but NEITHER statement ALONE is sufficient.
D. EACH statement ALONE issufficient.
E. Statements (1) and (2) TOGETHERare NOT sufficient.
为什么答案是C啊？