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

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; VectorPaint modulo-eight clocks: compare step-three coverage with two step-two cycles.

Sources checked on 15 November 2026: the official VectorPaint App Store listing and Apple public lookup 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 uses Euclid and Bézout relations for modular inverses. Seven,twenty-six and the card calculations are original; congruence is not ordinary real equality.