g10-12-03 · Additional mathematics
Why do remainders preserve the GCD?
Can replacing A by a remainder lose a common factor?
1 MIN · PREDICT ON PAPER
Start with A=(x−r)(x²+u), B=(x−r)(x+v); predict the final nonzero remainder.
Discuss or write first. No typed answer is required to continue.
Monic GCD: x −1
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
The last nonzero Euclidean remainder, made monic, is the GCD. Sometimes B already divides A. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: The displayed polynomials are nonzero; each remainder has smaller degree. The zero remainder terminates the algorithm.
Mathematical justification: If A=BQ+R, a divisor common to A,B divides R=A−BQ; conversely a divisor of B,R divides A. Repetition preserves all common divisors until termination.
1 MIN · EXIT CHECK
Why do A and B have the same GCD as B and A−B?
Check after answering
A common divisor of A,B divides A−B; a common divisor of B,A−B divides their sum A.
Teacher guidance and extension
Before: Polynomial division and factors
Ask: After every arrow, ask pupils to justify the same-common-divisors claim in both directions.
Watch for: A zero remainder is not itself the GCD; use the preceding nonzero divisor.
Challenge: Find a condition on u,v that makes the GCD quadratic rather than linear.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
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