Question

1. explain what the modulus operator % calculates.

Explanation:

The modulus operator % (also known as the remainder operator in some contexts) is used in mathematics and programming to find the remainder of a division operation between two integers. For example, when we divide an integer \( a \) by an integer \( b \) (where \( b
eq0 \)), the expression \( a\%b \) gives the remainder that is left over when \( a \) is divided by \( b \). Mathematically, if we express the division of \( a \) by \( b \) as \( a = q\times b + r \), where \( q \) is the quotient (the result of the integer division of \( a \) by \( b \)) and \( r \) is the remainder, then \( a\%b=r \), and the remainder \( r \) satisfies the condition \( 0\leq r < |b| \) (for positive divisors; the sign of the remainder can follow different conventions in some programming languages, but the core idea is about the remainder of integer division). For instance, \( 10\%3 = 1 \) because \( 10 = 3\times3+1 \), and the remainder is 1. In programming (e.g., in languages like Python, Java, C++), it operates on integer operands to return this remainder, which is useful for tasks like checking if a number is even or odd (e.g., \( n\%2 = 0 \) means \( n \) is even, \( n\%2 = 1 \) means \( n \) is odd), finding periodic patterns, or working with cyclic data structures.

Answer:

The modulus operator % calculates the remainder of the division of one integer (the dividend) by another non - zero integer (the divisor). Mathematically, for integers \( a \) (dividend) and \( b \) (divisor, \( b
eq0 \)), if \( a = q\times b + r \) (where \( q \) is the quotient from integer division and \( r \) is the remainder), then \( a\%b=r \), with \( 0\leq r < |b| \) (for positive divisors; sign conventions may vary in programming). It's used in tasks like checking parity (even/odd) and working with cyclic patterns.