# VectorPaint modular-inverse card: back-substitute to show seven times fifteen leaves one

Language: en
Canonical: [VectorPaint modular-inverse card: back-substitute to show seven times fifteen leaves one](<https://allluckly.cn/en/guides/vectorpaint-modular-inverse-seven-mod-twenty-six-back-substitution/>)
中文：[中文 HTML](<https://allluckly.cn/guides/vectorpaint-modular-inverse-seven-mod-twenty-six-back-substitution/>)
English: [English HTML](<https://allluckly.cn/en/guides/vectorpaint-modular-inverse-seven-mod-twenty-six-back-substitution/>)
Published: 2026-11-15T00:00:00+08:00
Modified: 2026-11-15T10:05:49+08:00
Author: Bison

Retain Euclidean remainders and coefficients, distinguishing modular inverse, reciprocal and noninvertible cases.

English translation of the [Chinese original](<https://allluckly.cn/guides/vectorpaint-modular-inverse-seven-mod-twenty-six-back-substitution/>), translated on 2026-11-15. Original publication dates, attribution, code and version notes are retained.

The inverse of seven modulo twenty-six is an integer giving remainder one after multiplication, not merely seven in a denominator. Draw a division/back-substitution card in VectorPaint, derive equivalent representatives -11 and 15, and multiply back. Use Euclid to find integer coefficients, retaining quotients, remainders and substitution signs throughout without built-in symbolic computation.

## State the multiplication-to-remainder-one target

Seek integer r with 7r−1 an integer multiple of 26, written 7r≡1 mod 26. Choose representatives 0..25 if desired; negatives may represent the same class. On an approximately 1800×1500 canvas separate division, substitution and verification. Label multiplicative inverse, not merely reciprocal. Decimal 1/7 solves a different real-number problem. Modulus 26 is fixed positive, not a quotient result. Manual conditions govern the diagram.

## Retain quotient and remainder in every division

Write 26=3×7+5,7=1×5+2,5=2×2+1,2=2×1+0. Each remainder is smaller than its divisor; the last nonzero remainder 1 gives gcd 1. Arrows pass divisor/remainder onward, not a factor tree. Preserve full products/sums and readable operators: do not replace the correct 5=2×2+1 with the incorrect 5=2+2. This has not yet produced the inverse: the final remainder one does not make r=1.

## Back-substitute from one

Start 1=5−2×2. Substitute 2=7−5 to obtain 1=5−2(7−5)=3×5−2×7. Substitute 5=26−3×7 to get 1=3×26−11×7. Preserve parentheses/signs: 2×7−5 differs from 2(7−5). Replace one remainder at a time and retain previous rows. Reading quotients backwards is insufficient. These are integer coefficients, not an assumed algebra simplifier.

## Normalise minus eleven into zero through twenty-five

Modulo 26, the term 3×26 vanishes, leaving 7×(-11)≡1. Add 26 to -11 to choose r=15; -11 is valid but outside the selected nonnegative range. Forty-one is another representative. Keep 15 as the main answer and alternatives in notes. Erasing the minus to make 11 fails: 77 leaves remainder 25. Normalisation changes representation, not modulus or the integer identity.

## Multiply back and check remainder one

7×15=105=4×26+1 verifies remainder one. Use equality for integer equations and congruence for modular relations, never 7×15=1. Independently,3×26−11×7=78−77=1. Controls 7×14=98 and 7×16=112 leave 20 and 8, not one. Candidate choice depends on residue, not proximity to 105 or drawn length.

## Keep nonexistence when coprimality fails

For a counterexample use 13 modulo 26: gcd=13, not one. Products of 13 have residues only 0 or 13, never 1, so no inverse exists. Do not invent r=2 or change the modulus to 27 just to fill every answer box. For coprime seven, one representative lies in 0..25; -11,15,41 belong to the same class, not three distinct modular inverses. Separate the counterexample and retain nonexistence as a valid outcome.

## Troubleshoot remainders, coefficients and notation

For r=11 recheck the final minus; for r=3 check whether the coefficient belongs to 26 rather than 7. Reduce residues into 0..25 without swapping quotient/remainder. Repair substitution arrows by the replaced remainder, not a simultaneous rewrite. Group parentheses/operators when moving text to preserve equations. Colours/arrows prove no number-theoretic relation, and a drawn division chain supplies no automatic coefficient or existence certification.

## Reopen from target to verification and counterexample

Save native work, export PNG and reopen. Follow 7r≡1 mod26 through four divisions and substitution to -11/15, then check 105’s remainder. Keep the no-inverse 13 control separate. Enlarge a tiny minus instead of deleting it. Reopen editable panels and retain equations/range. Explain coefficient origins, negative representatives and nonexistence, without confusing real 1/7, implying an automatic cryptographic tool or claiming a security test. Recalculate changed inputs.

## Related tutorials and sources

[VectorPaint Euclidean square tiling: split eighty-four by thirty into seven integer squares](<https://allluckly.cn/en/guides/vectorpaint-euclidean-square-tiling-eighty-four-by-thirty/>); [VectorPaint modulo-eight clocks: compare step-three coverage with two step-two cycles](<https://allluckly.cn/en/guides/vectorpaint-mod-eight-step-cycles/>).

Sources checked on 15 November 2026: the [official VectorPaint App Store listing](<https://apps.apple.com/cn/app/id1587272880>) and [Apple public lookup](<https://itunes.apple.com/lookup?id=1587272880&country=cn&entity=software&timestamp=20261115>) show version 1.0.8, requiring iOS / iPadOS 17.0 or later. Draw manually with published shapes, paths, nodes, layers and text. Conditions, scales and layouts are original exercise arrangements; check installed controls. Keep editable native work and PNG previews. SVG/PDF may retain any bitmap content used. External calculations and static diagrams are not new automatic proof, statistics or solver features. No completed device, physical-object or universal export testing is claimed.

Method: [Western University’s congruence chapter](<https://www.csd.uwo.ca/~abrandt5/teaching/DiscreteStructures/Chapter4/solve-congruences.html>) uses Euclid and Bézout relations for modular inverses. Seven,twenty-six and the card calculations are original; congruence is not ordinary real equality.