If x and y are consecutive positive integers such that x

Question

If x and y are consecutive positive integers such that x < y, which of the following statements is true without any exceptions?

I. (x + 1)(y - 1) = xy
II. (x + y)^2 leaves a remainder of 1 when divided by 8
III. The difference between the larger number and the sum of the remainders when x and y are divided by each other is 1.

A. I only
B. II only
C. III only
D. I and II only
E. II and III only

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sujoykrdatta · Answer

If x and y are consecutive positive integers such that x < y

=> y = x + 1

Which of the following statements is true without any exceptions?

I. (x + 1)(y - 1) = xy
=> (x + 1) (x + 1 - 1) = x * (x + 1)
=> (x + 1) * x = x * (x + 1)
=> TRUE

II. (x + y)^2 leaves a remainder of 1 when divided by 8
=> \((x + x + 1)^2\) = \((2x + 1)^2\) = \(4x^2 + 2*2x *1 + 1^2\) = \(4x^2 + 4x + 1\)
=> 4x * (x + 1) + 1 = 4*x*(x+1) + 1

Now, x*(x+1) is a product of two consecutive numbers
=> It will be even
=> 4*x(x+1) = 4*Even = Multiple of 8

=> Remainder of 4*x*(x+1) + 1 = 1 as 4*x*(x+1) is a multiple of 8
=> TRUE

III. The difference between the larger number and the sum of the remainders when x and y are divided by each other is 1.

y when divided by x will give 1 remainder as y = x+ 1 and x+1 divided by x will give 1 remainder
x when divided by y will give x remainder as x is smaller than y

=> Sum of remainders = 1 + x = y

Largest number out of x and y is y
=> Difference of largest number and sum of remainders = y - y = 0 ≠ 1
=> FALSE

So, Answer will be D
Hope it helps!

Watch the following video to MASTER Remainders

hr1212 · Answer

I. (x + 1)(y - 1) = xy

xy - x + y - 1 = xy
y - x = 1

Always true.

II. (x + y)^2 leaves a remainder of 1 when divided by 8

Let x be a and y be a + 1, then (a + a + 1)^2 = 4a^2 + 4a + 1 = 4a(a + 1) + 1

a*(a+1) should be divisible by 2 which means 4*a*(a + 1) should be divisible by 8, so (4a(a + 1) + 1)%8 should always have remainder of 1.

Always true.

III. The difference between the larger number and the sum of the remainders when x and y are divided by each other is 1.

x/y should always give remainder as x as x < y
y/x should always give remainder as 1 as y = x + 1

Sum = x + 1

Larger no. - sum of remainders = 1

y - (x + 1) = 1 (y - x = 1)
1 - 1 = 1
0 = 1

Not true.

IMO: D

AVMachine · Answer

x, y are consecutive, hence can be written as x, x+1; y = x+1;

I. (x + 1)(y - 1) = xy;
Simplify: xy + y - x - 1; put y=x+1;
x(x+1) + x+1 -x -1 = x^2 + x + x + 1 -x - 1 = x^2 + x = x(x+1) = xy

II. (x + y)^2 leaves a remainder of 1 when divided by 8 
Put y=x+1; (x + x +1)^2 = (2x+1)^2 = 4x^2 + 1 + 4x = 4x(x+1) + 1;

4x(x+1), since x and x+1 are two consecutive no. one out of them would be even; so 4x(x+1) leaves remainder as zero; Hence, ( 4x(x+1) + 1 )/8; will leaves 1 as remainder.

III. The difference between the larger number and the sum of the remainder when x and y are divided by each other is 1.

Consider, x=2; y=3

remainder(x/y) = 2; remainder(y/x) = 1; Sum of Remainders = 3;

3-3=0;

Ans: D

BrushMyQuant · Answer

If x and y are consecutive positive integers such that x < y

=> y = x + 1

Which of the following statements is true without any exceptions?

I. (x + 1)(y - 1) = xy 
=> (x + 1) (x + 1 - 1) = x * (x + 1)
=> (x + 1) * x = x * (x + 1)
 =>    TRUE

II. (x + y)^2 leaves a remainder of 1 when divided by 8 
=>  (x + x + 1)^2  =  (2x + 1)^2  =  4x^2 + 2*2x *1 + 1^2  =  4x^2 + 4x + 1 
 => 4x * (x + 1) + 1 = 4*x*(x+1) + 1

Now, x*(x+1) is a product of two consecutive numbers 
=> It will be even
=> 4*x(x+1) = 4*Even = Multiple of 8

=> Remainder of 4*x*(x+1) + 1 = 1 as 4*x*(x+1) is a multiple of 8
  =>    TRUE

III. The difference between the larger number and the sum of the remainders when x and y are divided by each other is 1.

y when divided by x will give 1 remainder as y = x+ 1 and x+1 divided by x will give 1 remainder
x when divided by y will give x remainder as x is smaller than y

=> Sum of remainders = 1 + x = y

Largest number out of x and y is y
=> Difference of largest number and sum of remainders = y - y = 0  ≠ 1
 =>   FALSE

So,  Answer will be D 
Hope it helps!

Watch the following video to MASTER Remainders

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

If x and y are consecutive positive integers such that x < y, which of

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