Question
Explanation
If a + b
= 11, then 2ab
= (a + b)² – (a² + b²) = 121 – 100 = 21. But 2ab
is an even number, whereas 21 is
odd. This is a contradiction. So (B) is
impossible. ©
Remarks
Short and sweet isn’t it? This uses the square-of-sum identity (a + b)² = a² + 2ab + b². I used the tactic
of assuming the answer is correct [H08] and showing that this leads to something nonsensical [H05]. So the original assumption must be
wrong. This is called “proof by contradiction” or reductio ad absurdum
(in Latin).
By the way, the correct
answer option is (E) from the Pythagorean Triplet 8² + 6²
= 10² with {a,
b} = {8, 6}. The question seems to be taken from some
Kangaroo mathematics competition.
H05. Work backwards
H08. Make suppositions
H09. Restate the problem in
another way
H13* Use Equation / write a
Mathematical Sentence
Suitable Levels
* Lower Secondary Mathematics competition
* GCE ‘O’ Level “Elementary” Mathematics (challenge)
* other syllabuses that involve whole
numbers and Pythagorean triplets
* any precocious or independent learner who
loves a challenge
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