Use integer witnesses rather than treating divisibility as unexplained symbolism.
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Try
Divisibility says that one integer is an integer multiple of another. If a bundle contains a objects, then b=ar and c=as describe whole numbers of bundles. Combining integer numbers of those bundles keeps a as a factor; division need not.
Predict
Predict whether the witnesses r and s must be equal. They do not: two quantities can share a divisor while containing different numbers of copies. In this unit a is a nonzero integer; a|b means there exists an integer r with b=ar.
Experiment
Choose a=3, b=12 and c=−6. Write their integer witnesses before computing 2b+c. Now change only c to 5: which premise fails? Use the changed case to separate a false premise from a false deduction. A theorem is not refuted by inputs that violate its assumptions.
Understand
Worked argument: if a|b and a|c, then a|(2b+c). By definition there are integers r,s with b=ar and c=as. Therefore 2b+c=2ar+as=a(2r+s). Since 2r+s is an integer, the definition proves the conclusion. The final sentence is essential: the coefficient must belong to the required domain.
An invalid argument writes b=ak and c=ak using the same k without justification. For a=3,b=6,c=9, the witnesses are 2 and 3. Using separate letters prevents an accidental claim that b=c.
Apply
Build your practice proofs in three moves: expand the assumptions into equations, rewrite the target, then verify that its coefficient is an integer. A numerical example can help you find the algebra but does not establish the general claim.
Continue
Depth: prove a divides ub+vc for any integers u,v. Then investigate why allowing rational u,v breaks the conclusion. Separately, some texts define divisibility even when a=0; that convention must be stated explicitly before extending this lesson’s nonzero-divisor domain.
Apply · explain · revise
Practice and reasoning
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Let a be a nonzero integer, b=4a and c=−3a. What integer k makes 3b−2c=ak?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
If a divides b and c, which conclusion is always valid for nonzero integer a?
Prove: if nonzero integer a divides integers b and c, then a divides 3b−2c.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A proof says a|b and a|c imply a|(b/c). Give an integer counterexample with c nonzero and explain the failed step.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
