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Question 376068: for all numbers x and y, let the operation _ be defined by x _ y = xy - y. If a and b are positive integers, which of the following can be equal to zero?
** the underscore or _ stands for a missing operation
1.) a _ b
2.) (a+b)_b
3. a_(a+b)
a.) one only
b.) two only
c.) three only
d.) one and two
e.) one and three
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website!
x _ y = xy - y
1.) a _ b
That means ab - b.
Let's set
ab - b = 0
b(a - 1) = 0
Using the zero factor property,
b = 0 or a - 1 = 0
a = 1
Since b is a positive integer we can rule out b = 0.
However if a = 1 then ab - b =0 when b is any positive integer.
So 1.) can be zero.
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2.) (a+b)_b
That means (a+b)b - b.
Let's set
(a+b)b - b = 0
Factor out b:
b[(a+b)-1] = 0
b[a+b-1] = 0
Using the zero-factor property again:
b = 0; a+b-1 = 0
a = 1-b
We must rule out b = 0 since b must be a positive integer.
We also must rule out a = 1-b since a positive integer b subtracted
from 1 canNOT be a positive integer.
So (2) canNOT be zero.
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3. a_(a+b)
That means a(a+b) - (a+b).
Let's set:
a(a+b) - (a+b) = 0
Factor out (a+b)
(a+b)(a-1) = 0
Using the zero-factor property again:
a+b = 0 ; a-1 = 0
a = -b; a = 1
We have to rule out a = -b because a and b are both positive integers,
but if a = 1, then
(1+b)(1-1) = (1+b)0 = 0, regardless of what positive integer b equals.
So 3.) can be zero
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So only
2.) (a+b)_b
cannot be zero. The other two can be zero.
Therefore the correct choice is
e.) one and three
Edwin
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