Questions: What is the order of 1 modulo 10?

Solution Steps

To find the order of 1 modulo 10, we need to determine the smallest positive integer \( k \) such that \( 1^k \equiv 1 \pmod{10} \). Since any power of 1 is 1, the order is 1.

The order of an integer \( a \) modulo \( n \) is the smallest positive integer \( k \) such that \( a^k \equiv 1 \pmod{n} \). In this case, we are looking for the order of \( 1 \) modulo \( 10 \).

We need to find the smallest \( k \) such that: \[ 1^k \equiv 1 \pmod{10} \] Since \( 1^k = 1 \) for any positive integer \( k \), it follows that: \[ 1 \equiv 1 \pmod{10} \] This holds true for \( k = 1 \).

Thus, the order of \( 1 \) modulo \( 10 \) is \( 1 \).

Final Answer