Divisibility by 3

[Uri]

If x and y are integers, is xy + 1 divisible by 3?
(1) When x is divided by 3, the remainder is 1.
(2) When y is divided by 9, the remainder is 8.

Please explain the logic.

OA: [spoiler](C)[/spoiler]
Source: GMAT Plus Book

[iamcste]

If x and y are integers, is xy + 1 divisible by 3?
(1) When x is divided by 3, the remainder is 1.
(2) When y is divided by 9, the remainder is 8.

Please explain the logic.

OA: [spoiler](C)[/spoiler]
Source: GMAT Plus Book

Stmt 1

X=Q1*3+1, Q1=positive integer- by definition
X=4,7,10,13,16,19,22,25,

we need Y Insufficient


Stmt 2
Y=Q2*9+8
17,26 ..

we need X Insufficient

Options left: C or E

Collectively,
X=4,7,10,13,16,19,22,25,
Y=17,26 ..

If I evaluate XY+1 ( 69, 163..) all are divisble by 3

Sufficient Choose C

[mals24]

question: Is xy+1 divisible by 3?

St 1: When x is divided by 3, the remainder is 1.

That is: X = 3A + 1

No information about y so statement is insuff.

St 2: When y is divided by 9, the remainder is 8.

That is: Y = 9B + 8

No information about x so statement is insuff.

Combining 1 and 2

X = 3A + 1
Y = 9B + 8

We need to find XY +1

XY + 1= (3A+1)(9B+8) + 1

Open the brackets: 27*AB + 3(8A) + 9B + 9
Since each term is multiplied by 3 atleast once, we can conclude that
XY +1 is divisible by 3.

So C

[Ian Stewart]

xy+1 is divisible by 3
means, xy+1 is odd
1 is odd, hence xy must be even to make xy+1 odd

Careful here - "xy + 1 is divisible by 3" does not mean "xy + 1 is odd". Many even numbers are divisible by 3: ...-12, -6, 0, 6, 12, 18, 24, 30, etc.

[iamcste]

xy+1 is divisible by 3
means, xy+1 is odd
1 is odd, hence xy must be even to make xy+1 odd

Careful here - "xy + 1 is divisible by 3" does not mean "xy + 1 is odd". Many even numbers are divisible by 3: ...-12, -6, 0, 6, 12, 18, 24, 30, etc.

Ian I have amended it already, thanks for pointing.

[austin]

If x and y are integers, is xy + 1 divisible by 3?
(1) When x is divided by 3, the remainder is 1.
(2) When y is divided by 9, the remainder is 8.

Statement 1: No info on y
Statement 2: No info on x

COMBINE: When y is divided by 9, the remainder is 8
=> when y is divided by 3, the remainder is 2

The remainder when xy is divided by 3 = r1 *r2 = r
where
r1 = remainder when x is divided by 3
r2 = remainder when y is divided by 3

r = 2*1 = 2

so when xy is divided by 3 remainder is 2 => xy+1 is divisible by 3

C