Number theory: exact arithmetic, structure and proof
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 240 minutes, as an unvalidated planning estimate. This collection contains 16 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Attempt the mixed problems before revealing guidance. State definitions, hypotheses and intermediate arguments, and identify any theorem you use without proving it.
These public tasks require human mathematical review. A saved response is not verified proof quality, a secure examination or evidence of universal course equivalence.
The suggested time is planning guidance; work is untimed. Preserve the initial attempt, consult targeted lessons, then revise your reasoning.
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0 of 16 tasks have recorded responses. No overall mastery score is assigned.
In(a+b+c)^7 find the coefficient of a³b⁴c⁰ and prove all non-pure coefficients vanish modulo7. Deduce10^7 modulo7 and justify whether10^6≡1 follows. A learner changes the degree and modulus to6 and claims every mixed coefficient still vanishes and every nonconstant binary word has six rotations. Repair both claims with one exact coefficient and one explicit word. Explain why a zero exponent is not an exception to the prime proof.
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.
Without using decimal approximations, classify (81/16)^(3/4), (81/16)^(2/3), and (81/16)^(−1/2) as rational or irrational, giving exact rational values when available. Explain the difference between the real cube root and fourth root of −8/27. Finally refute the claim that a rational product of two positive radicals forces both radicals rational, and repair the statement when one factor is known nonzero rational.
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.
For A={12,18,30}, compute gcd and lcm using prime coordinates. With base B=12 compute w_B of each operand, their gcd, lcm and product; identify which proposed minimum, maximum and sum identities hold here. Then give a necessary and sufficient prime-coordinate condition for a nonempty infinite positive-integer family to have a positive integer lcm. Apply it separately to {2^k:k≥1} and to the set of all primes.
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.
Find every positive n with φ(n)=8, then solve φ(n)=10 and φ(n)=9. Give a proof-based finite search for each even target. A proposed shortcut drops prime 2 because its first-power totient is one; explain exactly which answers this loses.
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 that x²+2y²=7z² has no nonzero integer solution and u²+2v²=63 has no rational solution. Give every reduction and the decreasing positive integer. Compare x²+3y²=7z²: determine why the same residue obstruction fails and give a genuine integer solution. Finally explain why checking triples in a bounded cube would not establish your nonexistence claim.
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.
A report claims that an empty root list from a capped odd-prime algorithm proves a nonsquare, and that one simple Hensel lift supplies all roots of a composite congruence. Repair both claims with exact evidence. For a changed target, find every root of x²≡4 modulo175 by starting modulo5 and7, lifting the5-branches to25, then using CRT. Include the signed quotient and derivative for each lift and prove completeness. Contrast this with starting root0 modulo5 for x²≡5 modulo25. Finally explain the status of x≡1(mod6),x≡4(mod9) under a coprime-only implementation.
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.
Use Euclid to find gcd(391,299), express it as a linear combination, and solve 391x+299y=46 in all integers. Explain why the same left side cannot equal 45. Prove the prime-divisor lemma used in uniqueness of factorization.
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.
Solve 18x≡30 (mod 84), listing all residue classes modulo 84. Solve x≡2 (mod 9), x≡5 (mod 14). Decide whether x≡1 (mod 6), x≡2 (mod 9) is solvable. Explain why cancelling a factor does not preserve the original modulus.
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 2 is a primitive root modulo 19, then modulo 19². State and justify the lifting criterion you use. Explain why Euler’s theorem alone does not identify an element’s order.
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.
Starting from Σd|n φ(d)=n, derive a formula for φ using Möbius inversion and evaluate it at n=72. Prove the inversion identity by exchanging finite divisor sums. Then find g(12) when f(n)=n² and f(n)=Σd|n g(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.
Decide whether 29 is a square modulo 43 using quadratic reciprocity and supplement laws, then state the corresponding Euler-criterion value. Solve x²≡9 (mod 65) completely, and explain why a prime-field two-root bound does not hold for this composite modulus.
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.
Lift both roots of x²≡14 (mod 5) to roots modulo 25 and 125. Explain why each lift is unique. Give the order-decrease invariant behind Tonelli–Shanks and explain why it is inappropriate to apply that prime-modulus algorithm directly to modulus 125.
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.
Compute the first four convergents of √7. Identify a Pell solution and prove that powers of its associated unit give integer solutions. Prove it is the least positive nontrivial solution without relying on the finite search alone.
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.
Parametrize rational points on x²+y²=1 from (−1,0), and use t=3/4 to obtain a primitive integer right triangle. Determine whether 245 is a sum of two integer squares. Give a four-square representation of 23 and explain the strict-decrease requirement in an infinite-descent proof.
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.
Evaluate Φ6(5) and explain the multiplicative-order information its prime factor gives. Derive Wilson’s congruence for a prime using inverse pairs. For toy RSA p=7,q=17,e=5, find d modulo φ(pq), encrypt message 2 and recover it, including an argument for messages not coprime to pq.
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.
For real s>1, justify ζ(s)=∏p(1−p^(−s))⁻¹ and Σn≥1 μ(n)n^(−s)=1/ζ(s). Explain precisely why the same rearrangement argument does not prove an identity at s=1 or in a conditionally convergent region.
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.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
In(a+b+c)^7 find the coefficient of a³b⁴c⁰ and prove all non-pure coefficients vanish modulo7. Deduce10^7 modulo7 and justify whether10^6≡1 follows. A learner changes the degree and modulus to6 and claims every mixed coefficient still vanishes and every nonconstant binary word has six rotations. Repair both claims with one exact coefficient and one explicit word. Explain why a zero exponent is not an exception to the prime proof.
Without using decimal approximations, classify (81/16)^(3/4), (81/16)^(2/3), and (81/16)^(−1/2) as rational or irrational, giving exact rational values when available. Explain the difference between the real cube root and fourth root of −8/27. Finally refute the claim that a rational product of two positive radicals forces both radicals rational, and repair the statement when one factor is known nonzero rational.
For A={12,18,30}, compute gcd and lcm using prime coordinates. With base B=12 compute w_B of each operand, their gcd, lcm and product; identify which proposed minimum, maximum and sum identities hold here. Then give a necessary and sufficient prime-coordinate condition for a nonempty infinite positive-integer family to have a positive integer lcm. Apply it separately to {2^k:k≥1} and to the set of all primes.
Find every positive n with φ(n)=8, then solve φ(n)=10 and φ(n)=9. Give a proof-based finite search for each even target. A proposed shortcut drops prime 2 because its first-power totient is one; explain exactly which answers this loses.
Prove that x²+2y²=7z² has no nonzero integer solution and u²+2v²=63 has no rational solution. Give every reduction and the decreasing positive integer. Compare x²+3y²=7z²: determine why the same residue obstruction fails and give a genuine integer solution. Finally explain why checking triples in a bounded cube would not establish your nonexistence claim.
A report claims that an empty root list from a capped odd-prime algorithm proves a nonsquare, and that one simple Hensel lift supplies all roots of a composite congruence. Repair both claims with exact evidence. For a changed target, find every root of x²≡4 modulo175 by starting modulo5 and7, lifting the5-branches to25, then using CRT. Include the signed quotient and derivative for each lift and prove completeness. Contrast this with starting root0 modulo5 for x²≡5 modulo25. Finally explain the status of x≡1(mod6),x≡4(mod9) under a coprime-only implementation.
Use Euclid to find gcd(391,299), express it as a linear combination, and solve 391x+299y=46 in all integers. Explain why the same left side cannot equal 45. Prove the prime-divisor lemma used in uniqueness of factorization.
Prove 2 is a primitive root modulo 19, then modulo 19². State and justify the lifting criterion you use. Explain why Euler’s theorem alone does not identify an element’s order.
Starting from Σd|n φ(d)=n, derive a formula for φ using Möbius inversion and evaluate it at n=72. Prove the inversion identity by exchanging finite divisor sums. Then find g(12) when f(n)=n² and f(n)=Σd|n g(d).
Decide whether 29 is a square modulo 43 using quadratic reciprocity and supplement laws, then state the corresponding Euler-criterion value. Solve x²≡9 (mod 65) completely, and explain why a prime-field two-root bound does not hold for this composite modulus.
Lift both roots of x²≡14 (mod 5) to roots modulo 25 and 125. Explain why each lift is unique. Give the order-decrease invariant behind Tonelli–Shanks and explain why it is inappropriate to apply that prime-modulus algorithm directly to modulus 125.
Compute the first four convergents of √7. Identify a Pell solution and prove that powers of its associated unit give integer solutions. Prove it is the least positive nontrivial solution without relying on the finite search alone.
Parametrize rational points on x²+y²=1 from (−1,0), and use t=3/4 to obtain a primitive integer right triangle. Determine whether 245 is a sum of two integer squares. Give a four-square representation of 23 and explain the strict-decrease requirement in an infinite-descent proof.
Evaluate Φ6(5) and explain the multiplicative-order information its prime factor gives. Derive Wilson’s congruence for a prime using inverse pairs. For toy RSA p=7,q=17,e=5, find d modulo φ(pq), encrypt message 2 and recover it, including an argument for messages not coprime to pq.
For real s>1, justify ζ(s)=∏p(1−p^(−s))⁻¹ and Σn≥1 μ(n)n^(−s)=1/ζ(s). Explain precisely why the same rearrangement argument does not prove an identity at s=1 or in a conditionally convergent region.
