algebra

[prachi18oct]

If a, b are nonzero integers, what is the value of ab?

(1) a = ab - 1
(2) a + b = 3

[theCEO]

If a, b are nonzero integers, what is the value of ab?

(1) a = ab - 1
(2) a + b = 3

(1)
a = ab - 1
ab = a + 1
a can be any value; statement is insufficient

(2)
a + b = 3
a and b can have any value, so statement is insufficient

(combining equations)
a+b = 3 ---- from (2)
b = 3 -a
ab = a+1 ---- from (1)
a (3-a) = a + 1
3a -a^2 -a -1 = 0
-a^2 + 2a -1 = 0
a^2 - 2a + 1 = 0
(a - 1)(a-1) = 0
a-1 = 0
a = 1

b = 3 -a
b = 3 - 1 = 2

ab = 2 x 1 = 2
statement is sufficent
Ans = C

[Ian Stewart]

Statement 2 is not sufficient, because a and b might be, among other possibilities, 2 and 1, or 4 and -1, and ab can have different values.

For Statement 1, we have:

a = ab - 1
1 = ab - a
1 = a(b-1)

So we know that a and b-1 are integers which multiply together to give 1. That can only happen if they're both equal to 1, or both equal to -1. But b-1 cannot equal -1, because that would mean b=0, and we know b is nonzero. So the only way the equation can be true is if a=1 and b-1 = 1, so b=2, and ab must be 2.

So Statement 1 is sufficient and the answer is A.

[theCEO]

Statement 2 is not sufficient, because a and b might be, among other possibilities, 2 and 1, or 4 and -1, and ab can have different values.

For Statement 1, we have:

a = ab - 1
1 = ab - a
1 = a(b-1)

So we know that a and b-1 are integers which multiply together to give 1. That can only happen if they're both equal to 1, or both equal to -1. But b-1 cannot equal -1, because that would mean b=0, and we know b is nonzero. So the only way the equation can be true is if a=1 and b-1 = 1, so b=2, and ab must be 2.

So Statement 1 is sufficient and the answer is A.

Thanks Ian. It's funny how I overlooked the word "integer".