Read congruences in Book Isle: choose remainder representatives before cancelling factors

Minus seven and five are different integers but congruent modulo twelve. Read original remainder cards in Book Isle, declare nonnegative representatives and test divisibility of differences. Check remainder classes, ordinary equality and modular rules. Calculate manually without assuming a remainder calculator or programming interpreter in the app.

Declare a positive modulus and representative range

Fix modulus twelve and choose representatives zero through eleven. Record source and range; including twelve would create thirteen positions. Twelve belongs with zero but zero is selected here. These are integers, not undefined decimal or negative-modulus extensions. A clock sketch may help, with an explicit mapping between its twelve label and representative zero rather than calling twelve a permitted final remainder.

Express minus seven using quotient minus one

Write minus seven as twelve times minus one plus five, with five in the selected range. Truncating the quotient toward zero would leave minus seven, outside that range. Check identity and convention instead of treating a tool output as universal. Programming remainder conventions are outside this paper exercise. Keep quotient, remainder and integer distinct; minus one is the quotient, not the representative.

Check congruence through divisible differences

Twenty-nine is twelve times two plus five, and its difference from minus seven is thirty-six, divisible by twelve. Record -7 ≡ 29 (mod 12) without replacing congruence by false ordinary equality. Five and seventeen are additional class members; choosing one representative does not erase them. Separate equality and congruence and enlarge an unclear triple-bar symbol rather than guess a double bar.

Recalculate addition and multiplication consistently

Minus seven plus eight is one; replacing minus seven by five gives thirteen, also representative one. Minus seven times three is minus twenty-one, twelve times minus two plus three; five times three gives fifteen, again representative three. Agreement uses the same modulus, not equality of every intermediate integer. Preserve direct and representative-first routes. Changing to modulus ten requires recalculating rather than reusing five.

Use a declared twenty-four-hour model separately

A separate twenty-four-hour cycle has no date or time-zone changes. Twenty-three plus seventeen is forty, one full twenty-four plus sixteen: representative sixteen and one wrap. This is not modulus twelve, so reducing sixteen to four changes systems. Real scheduling needs date, zone and rules absent here. If elapsed amount matters, retain cumulative forty and the one-wrap decomposition; a representative alone discards information.

Do not cancel factors as in ordinary equality

Two times one and two times seven are two/fourteen, congruent modulo twelve, while one/seven are not: difference six is not divisible by twelve. Thus cancelling two does not preserve that modulus automatically. Record the counterexample rather than transfer real-number division rules. Additional invertibility conditions are needed; this exercise does not claim every nonzero residue has an inverse or require automatic inversion.

Check negative representatives and mixed moduli

For a final minus seven, check the zero-to-eleven convention. Thirteen reduces by subtracting twelve, not deleting a digit to make three. Four on the twenty-four-hour card suggests the wrong modulus. Restore congruence and modulus rather than call it approximately equality. Reconnect wrapped conditions and enlarge unclear symbols. Share original identities and conventions without invented language outputs or claims that the app executed code.

Reopen with relations and a cancellation counterexample

Reopen both cards and distinguish moduli twelve/twenty-four. Recalculate representatives five, one, three and sixteen, retaining divisible difference thirty-six, false ordinary equality and the noncancellation example. Explain retained versus discarded information. Reassess changed moduli/conventions; bookmarks only locate sources. Another reader should verify each step by divisibility rather than treat triple bars as decoration or one paper result as every software convention.

Related guides and sources

Book Isle Euclidean algorithm: trace divisors, remainders and the stopping condition; Read inequalities in Book Isle: check direction and endpoints when dividing by a negative.

Checked on November 15, 2026: Apple public information lists Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. See the official App Store for published capabilities; verify controls in the installed version.

Method reference: UBC PLP: congruence modulo n. Materials and numbers are original exercises. Record external reports, paper reasoning and actual reading observations separately, without inventing missing conditions. Sources do not establish new automatic analysis by the app. No step-by-step physical-device, private-server or all-output-environment testing is claimed.