Three positive integers a, b and c leave the same remainder y

Question

Three positive integers a, b and c leave the same remainder y when divided by x, where x is a prime number greater than 3 and y is an integer greater than 1. What is the remainder when x is divided by y?

(1) c + 3 when divided by x yields an integer

(2) a + x - 2 when divided by x yields an integer

(1) c + 3 when divided by x yields an integer

(2) a + x - 2 when divided by x yields an integer

WiziusCareers1 · Answer

Three positive integers a, b and c leave the same remainder y when divided by x, where x is a prime number greater than 3 and y is an integer greater than 1. What is the remainder when x is divided by y?

(1) c + 3 when divided by x yields an integer

(2) a + x - 2 when divided by x yields an integer

(1) c + 3 when divided by x yields an integer

(2) a + x - 2 when divided by x yields an integer

Pre-thinking on the question
x is an odd prime number
y is anything greater than 1.
Nothing much can be said from this information.

Considering Statement (1) alone:
Let c = 18, x = 7 => y = 4
So, when x is divided by y, the remainder is 3.

Let c = 12, x = 5 => y = 2
So, when x is divided by y, the remainder is 1.

INSUFFICIENT

Considering statement (2) alone:
Let a = 12, x = 5 => y = 2
So, when x is divided by y, the remainder is 1.

Let a = 5, x = 3 => y = 2
So, when x is divided by y, the remainder is 1.

Let a = 16, x = 7 => y = 2
So, when x is divided by y, the remainder is 1.

Clearly, the statement means that each of a, b, and c are 2 more than a multiple of x.
SUFFICIENT.

The answer is (B).

nick1816 · Answer

a=y mod x
b= y mod x
c= y mod x
x>y and x>3

Statement 1
c+3=0 mod x
(y+3) mod x= 0 mod x
y=-3 or x-3

x=3 mod y when y>3
x= 1 mod y when y<3

insufficient

Statement 2
a + x - 2= 0 mod x
a-2= 0 mod x
(y-2) mod x= 0 mod x
y=2

As x is an odd number, x=1 mod 2

Sufficient

IanStewart · Answer

Statement 1 tells us c is 3 less than some multiple of x. To find a remainder, we'd want to know how much greater c is than a multiple of x, so Statement 1 is not very useful.

If, as Statement 2 tells us, a+x-2 is divisible by x, then a-2 must be divisible by x as well (since subtracting x from a multiple of x will always give us another multiple of x). And if a-2 is divisible by x, then a must be 2 greater than an exact multiple of x, which is another way of saying (since x > 2) "the remainder is 2 when we divide a by x". So y = 2, and when we divide the odd prime x by 2, we must get a remainder of 1, and Statement 2 is sufficient alone.

I have no idea what the unknown "b" has to do with anything.

PyjamaScientist · Answer

I will just add to this brilliant explanation by Ian.
Why statement 2 is sufficient is as following-

Given, a= k1*x + y --------- (1)
Statement 2 gives: 
a + x - 2 = k2*x
Rearranging,
a - 2 = (k2 - 1)*x
a - 2 = Kx
Substitute value of a from one above,
k1*x + y - 2 = Kx
Rearrange,
y = x(K - k1) + 2
y = x*C + 2. \Here, K, k1, k2, C are all constants.

Hence, this is the form:
Dividend = Divisor * Quotient + Remainder

Where, Remainder = 2. Your answer.

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Three positive integers a, b and c leave the same remainder y

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Three positive integers a, b and c leave the same remainder y

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