Do a table of modulo 6 for me in division ➗
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Step-by-step solution
3 stepsLet's create a division table modulo 6. In modular arithmetic, division a ÷ b (mod n) is defined as finding a number x such that b · x ≡ a (mod n). This is equivalent to multiplying 'a' by the multiplicative inverse of 'b', denoted b⁻¹, where b · b⁻¹ ≡ 1 (mod n). A multiplicative inverse b⁻¹ exists only if the greatest common divisor (gcd) of b and n is 1.
Identify numbers with multiplicative inverses modulo 6.
For modulo 6, we consider numbers in the set {0, 1, 2, 3, 4, 5}.
- For b = 0: Division by zero is undefined.
- For b = 1: gcd(1, 6) = 1. The inverse is 1⁻¹ = 1 (since 1 × 1 = 1 mod 6).
- For b = 2: gcd(2, 6) = 2 ≠ 1. No multiplicative inverse exists.
- For b = 3: gcd(3, 6) = 3 ≠ 1. No multiplicative inverse exists.
- For b = 4: gcd(4, 6) = 2 ≠ 1. No multiplicative inverse exists.
- For b = 5: gcd(5, 6) = 1. The inverse is 5⁻¹ = 5 (since 5 × 5 = 25 ≡ 1 mod 6).
Calculate the division results.
For each 'a' (dividend) and 'b' (divisor), we calculate a ÷ b (mod 6). If b⁻¹ exists, the result is (a × b⁻¹) mod 6. If b⁻¹ does not exist or b = 0, the division is undefined or has no unique solution. We will denote these cases as "N/A".
Construct the division table modulo 6.
The rows represent the dividend 'a', and the columns represent the divisor 'b'.
Division Table Modulo 6
a \ b 0 1 2 3 4 5 0 N/A 0 N/A N/A N/A 0 1 N/A 1 N/A N/A N/A 5 2 N/A 2 N/A N/A N/A 4 3 N/A 3 N/A N/A N/A 3 4 N/A 4 N/A N/A N/A 2 5 N/A 5 N/A N/A N/A 1 Explanation of "N/A" entries:
- The column for b = 0 is N/A because division by zero is undefined.
- The columns for b = 2, 3, and 4 are N/A because these numbers do not have a unique multiplicative inverse modulo 6 (their gcd with 6 is not 1).
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