Calculator Form
Plotly Graph
The chart maps each residue to its inverse whenever one exists. Nonunits appear as missing points when the full residue mode is selected.
Z15 Inverse Lookup Table
| a | gcd(a,15) | Unit? | Inverse | Check |
|---|---|---|---|---|
| 0 | 15 | No | None | Not invertible |
| 1 | 1 | Yes | 1 | 1 mod 15 |
| 2 | 1 | Yes | 8 | 1 mod 15 |
| 3 | 3 | No | None | Not invertible |
| 4 | 1 | Yes | 4 | 1 mod 15 |
| 5 | 5 | No | None | Not invertible |
| 6 | 3 | No | None | Not invertible |
| 7 | 1 | Yes | 13 | 1 mod 15 |
| 8 | 1 | Yes | 2 | 1 mod 15 |
| 9 | 3 | No | None | Not invertible |
| 10 | 5 | No | None | Not invertible |
| 11 | 1 | Yes | 11 | 1 mod 15 |
| 12 | 3 | No | None | Not invertible |
| 13 | 1 | Yes | 7 | 1 mod 15 |
| 14 | 1 | Yes | 14 | 1 mod 15 |
Example Data Table
| Input a | Normalized in Z15 | gcd(a,15) | Inverse | Reason |
|---|---|---|---|---|
| 2 | 2 | 1 | 8 | 2 × 8 = 16 ≡ 1 mod 15 |
| 4 | 4 | 1 | 4 | 4 × 4 = 16 ≡ 1 mod 15 |
| 7 | 7 | 1 | 13 | 7 × 13 = 91 ≡ 1 mod 15 |
| 10 | 10 | 5 | None | Not a unit because gcd(10,15) ≠ 1 |
| -2 | 13 | 1 | 7 | 13 × 7 = 91 ≡ 1 mod 15 |
Formula Used
An element a in Z15 has a multiplicative inverse exactly when gcd(a, 15) = 1.
If gcd(a, 15) = 1, then integers x and y exist such that ax + 15y = 1.
Reducing x modulo 15 gives the inverse: a-1 ≡ x (mod 15).
The invertible residues are 1, 2, 4, 7, 8, 11, 13, and 14.
How to Use This Calculator
- Enter any whole number in the input field.
- The calculator automatically reduces it into Z15.
- Choose canonical or balanced output display.
- Select whether the graph should show units only or all residues.
- Enable Euclidean steps if you want the working process.
- Press Calculate Inverse to see the result above the form.
- Use the CSV or PDF buttons to export the computed information.
Frequently Asked Questions
1) What is a multiplicative inverse in Z15?
It is a residue b such that a × b ≡ 1 mod 15. Only unit elements have inverses, so the gcd test decides whether an inverse exists.
2) Why do some numbers have no inverse?
If a number shares a factor with 15, then gcd(a,15) is not 1. In that case, no product with that residue can reduce to 1 modulo 15.
3) Which residues are invertible in Z15?
The units are 1, 2, 4, 7, 8, 11, 13, and 14. Each of these is relatively prime to 15 and has exactly one inverse modulo 15.
4) Why does the calculator normalize negative inputs?
Modular arithmetic works with residue classes. A value like -2 belongs to the same class as 13 in Z15, so the calculator converts it to its standard residue.
5) What does the balanced display option do?
Balanced display rewrites residues using nearby negative representatives. For example, 13 can be shown as -2, which some students find easier during hand calculations.
6) How is the inverse actually computed?
The calculator uses the extended Euclidean algorithm. It finds integers x and y with ax + 15y = 1, then reduces x modulo 15 to obtain the inverse.
7) Why are 4 and 14 their own inverses?
Because 4 × 4 = 16 and 14 × 14 = 196. Both leave remainder 1 when divided by 15, so each residue is its own multiplicative inverse.
8) What is the benefit of the graph?
The graph helps you see patterns among units in Z15. It quickly shows which values are invertible and where inverse pairs or self-inverse residues occur.