If x is a positive integer, what is the remainder when x is divided by

Question

If x is a positive integer, what is the remainder when x is divided by 9?

(1) The remainder when 3x is divided by 27 is 24
(2) The remainder when 4x is divided by 18 is 14

datrose24 · Answer

(1): Because 3x divided by 27 leaves 24
-> 3x = 27k + 24 (k is an integer)
-> x = 9k +8
-> x divided by 9 remainder 8
-> (1) can solve

(2): 4x divided by 18 leaves 14
-> 4x = 18k +14
-> 2x = 9k + 7 = 9(k-1) +16
Since 16 is even and 2x is even, 9(k-1) is even -> 9.(k-1)/2 is divisible by 9
-> x = 9.(k-1)/2 + 8
-> x divided by 9 remainder 8
-> (2) can slove
my answer is D.

Kinshook · Answer

Asked: If x is a positive integer, what is the remainder when x is divided by 9?

(1) The remainder when 3x is divided by 27 is 24
3x = 27k + 24; where k is an integer.
x = 9k + 8
The remainder when x is divided by 9 = 8
SUFFICIENT

(2) The remainder when 4x is divided by 18 is 14
4x = 18k + 14; where k is an integer.
2x = 9k + 7
Sicne 2x is even and 7 is odd, 9k is odd. x is an odd multiple of 9. Let k = 2m + 1
2x = 9(2m+1) + 7
2x = 18m + 16
x = 9m + 8
The remainder when x is divided by 9 = 8
SUFFICIENT

IMO D

zhanbo · Answer

The approach below may not be optimal. But I was able to pick the right answer rather confidently after 02:39.

(1) Only 
Start with 3x=24 => x = 8. Note the reminder when x is divided by 9 is 8.
Next possible value for 3x is 3x=24+27=51 => x = 17. Note the reminder when x is divided by 9 is 8.
3x=78 => x=26. Note the reminder when x is divided by 9 is 8.
It is fairly certain to believe (1) is sufficient. The answer is 8.

(2) Only 
Start with 4x=14. Not possible because x is an integer. 
Try 4x=14+18 = 32 => x=8
Try 4x=32 +18*2 = 68 => x = 17
Exactly the same sequence as (1). 
It is fairly certain to believe (1) is sufficient. The answer is 8.

BrushMyQuant · Answer

If x is a positive integer, what is the remainder when x is divided by 9

STAT 1: The remainder when 3x is divided by 27 is 24

Theory: Dividend = Divisor*Quotient + Remainder

3x -> Dividend
27 -> Divisor
a -> Quotient (Assume)
24 -> Remainders
=> 3x = 27*a + 24 = 27a + 24

Dividing both the sides by 3 we get
x =  \frac{27a + 24}{3}  = 9a + 8

=> x when divided by 9 gives 8 as remainder
=>   SUFFICIENT

STAT 2: The remainder when 4x is divided by 18 is 14

Assume quotient = b
=> 4x = 18*b + 14 = 18b + 14
=> 2x = 9b + 7 = 9b + 9 - 2 = 9*(b+1) - 2

Since, 2x and 2 both are even => 9*(b+1) is also even
=> b + 1 is even = 2c (Assume)

=> 2x = 9*2c - 2
=> x = 9c - 1

=> x when divided by 9 gives (9-1) = 8 remainder
=>   SUFFICIENT

So,  Answer will be D 
Hope it helps!

Watch the following video to learn the Basics of Remainders

https://www.youtube.com/watch?v=P0S6j2uTaj4

bumpbot · Answer

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If x is a positive integer, what is the remainder when x is divided by

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