Question

If X = 11 and Y = 3, then X mod Y = (X + aY) mod Y holds

• A. Only for even integral values of a
• B. Only for odd integral values of a
• C. for all integral values of a
• D. for a = 0 only

Hint:

For any integers X, Y, and 'a', the expression `(X + aY) mod Y` will always be equal to `X mod Y`. This is because adding a multiple of the modulus (Y) does not change the remainder.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question tests a fundamental property of modular arithmetic. The expression `A mod B` gives the remainder when A is divided by B. The property in question is `(A + kB) mod B = A mod B`, where k is any integer. This is because `kB` is a multiple of `B`, and adding a multiple of the divisor does not change the remainder.
Step 2: Key Formula or Approach:
We need to evaluate both sides of the equation `X mod Y = (X + aY) mod Y` with the given values and see for which values of 'a' the equality holds.
The key property of modular arithmetic is: \[ (A + B) \mod N = ((A \mod N) + (B \mod N)) \mod N \] Step 3: Detailed Explanation:
Given values are X = 11 and Y = 3.
First, let's calculate the left-hand side (LHS) of the equation: \[ \text{LHS} = X \mod Y = 11 \mod 3 \] When 11 is divided by 3, the quotient is 3 and the remainder is 2. (Since \(11 = 3 \times 3 + 2\)).
So, LHS = 2.
Now, let's analyze the right-hand side (RHS) of the equation: \[ \text{RHS} = (X + aY) \mod Y = (11 + a \times 3) \mod 3 \] Here, 'a' is an integer. The term `3a` is always a multiple of 3, regardless of whether 'a' is even, odd, positive, negative, or zero.
According to the property of modular arithmetic: \[ (11 + 3a) \mod 3 = ((11 \mod 3) + (3a \mod 3)) \mod 3 \] We know that `11 mod 3 = 2`.
And since `3a` is a multiple of 3, the remainder when `3a` is divided by 3 is always 0. \[ 3a \mod 3 = 0 \] Substituting these values back: \[ \text{RHS} = (2 + 0) \mod 3 = 2 \mod 3 = 2 \] Step 4: Final Answer:
We see that LHS = 2 and RHS = 2. The equality holds true for any integer value of 'a'. Therefore, the statement is true for all integral values of 'a'.