Modulo Operation
Definition of the Modulo Operation
For two integers \(a\) (the dividend) and \(b\) (the divisor, \(b \neq 0\)), the modulo operation \(a \% b\) gives the remainder \(r\) of the division of \(a\) by \(b\). This can be written as:
\(a = b \cdot q + r\)
where:
- \(q\) is the quotient (an integer),
- \(r\) is the remainder,
- \(0 \le r < |b|\).
Applying the Definition
To prove that \(1 \% 10 = 1\) in mathematics, we use the formal definition of the modulo operation.
Given \(a = 1\) and \(b = 10\):
1. Division:
\(1 \div 10 = 0\)
The integer part of the quotient is \(0\), because \(10\) goes into \(1\) zero times.
2. Remainder:
To find the remainder, we use the relationship:
\(1 = 10 \cdot 0 + r\)
Solving for \(r\):
\(1 = 0 + r\)
\(r = 1\)
Verification
The remainder \(r\) must satisfy the condition:
\(0 \le r < 10\)
Here:
\(0 \le 1 < 10\)
The condition is satisfied, confirming that \(r = 1\).
Conclusion
Therefore:
\(1 \% 10 = 1\)
This proves that the remainder when \(1\) is divided by \(10\) is \(1\), consistent with the definition of the modulo operation in mathematics.
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