Dividing 23 by 5 gives 23=4·5+3. Dividing −23 by 5 gives −23=(−5)·5+2 when the remainder must lie between zero and four. A calculator that truncates toward zero may display a different quotient convention. Write the equality and the remainder range together; neither alone identifies the division theorem being used.
Predict
Predict whether a|b means a is smaller than b. It means that b=ak for some integer k, so signs and zero need care. For example, 7 divides zero and −3 divides 12. Zero divides only zero under this definition, since 0·k=0. Many later gcd statements exclude the pair (0, 0) so that a positive greatest common divisor can be chosen without ambiguity.
Experiment
Inspect a short remainder table.
Integer n
Divisor d
Quotient q
Remainder r
23
5
4
3
−23
5
−5
2
25
5
5
0
Check n=dq+r in every row. Then produce all integers with remainder two modulo five. The list extends in both directions; a residue class is not merely one nonnegative representative.
Understand
The well-ordering principle says every nonempty set of nonnegative integers has a least element. For d>0, consider nonnegative integers of the form n−dq with q integral. The set is nonempty: choosing q sufficiently negative makes the expression nonnegative. Let r be its least member. If r≥d, then r−d is a smaller nonnegative member, a contradiction. Hence n=dq+r with 0≤r< d. For uniqueness, subtract two such equations. A multiple of d equal to a difference of remainders strictly between −d and d must be zero. This proof explains both existence and the importance of the remainder interval.
Apply
Divisibility is an algebraic statement with a witness, so proofs should expose that witness. If a divides b and c, it divides ub+vc for any integers u, v. This closure under combinations drives Bézout identities and modular cancellation later. Strong induction is a complementary order tool: to prove a statement for n, one may assume it for every smaller relevant positive integer. It proves existence of prime factorizations by splitting a composite number into two smaller factors. It does not prove uniqueness; that requires a further divisibility lemma. Keeping these claims separate prevents a circular proof of the fundamental theorem of arithmetic.
Keep the role of zero explicit: every nonzero integer divides zero, while zero divides only zero under the integer-witness definition. Division with a remainder requires a positive divisor, so it never divides by zero.
Continue
Use negative inputs and zero cases when checking an algorithm, not only familiar positive examples. A written division implementation should state its quotient convention and verify the invariant n=dq+r with the allowed remainder. The next lesson repeatedly applies division to compute a gcd and recovers an integer combination of the inputs. Proofs in this unit use exact integers; finite machine arithmetic needs overflow handling, and a numerical trace is evidence for that input rather than a proof for every integer.
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Find the remainder when −29 is divided by 6, requiring 0≤r<6.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
In that same division, find the quotient.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Prove a|b and b|c imply a|c.
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.
Prove a common divisor of b, c divides every integer combination ub+vc.
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.
Prove uniqueness of q, r in n=dq+r with d>0 and 0≤r<d.
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.
Prove every integer n≥2 is a product of primes using strong induction.
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.
