Practice question · Put in order
Order the steps of solving 4x is congruent to 6 mod 7.
- Check gcd(4, 7) = 1, so a unique solution exists
- Find the inverse of 4 mod 7: it is 2, since 4 x 2 = 8 = 1 mod 7
- Reduce to get x congruent to 5 mod 7
- Multiply both sides by 2, giving x congruent to 2 x 6 = 12 mod 7
Hints
- Establish that a solution exists before hunting for it.
- Multiplying by the inverse is what replaces 'dividing by 4'.
Show the answer
- Check gcd(4, 7) = 1, so a unique solution exists
- Find the inverse of 4 mod 7: it is 2, since 4 x 2 = 8 = 1 mod 7
- Multiply both sides by 2, giving x congruent to 2 x 6 = 12 mod 7
- Reduce to get x congruent to 5 mod 7
Why
Coprimality first, then the inverse, then multiply, then reduce. Check the answer: 4 x 5 = 20 = 7(2) + 6. Skipping the coprimality check is what leads students to 'divide' in cases where no inverse exists at all.
Practise Congruences and Applications
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