12 multiple-choice questions, progressively harder.
The right triangle shown has side lengths 133133133, 156156156, and 205205205. What makes this triple primitive?
Solution
Correct answer: C
The legs are coprime, since 133=7⋅19133=7\cdot19133=7⋅19 and the prime factors of 156156156 are 222, 333, and 131313.
A composite hypotenuse is allowed.
Which triple does the pair (8,1)(8,1)(8,1) generate, with the hypotenuse last?
Correct answer: A
Use the square difference, the doubled product, and the square sum.
A triangle with sides 171717, 144144144, and 145145145 is enlarged. Its shortest side becomes 858585. What is the new hypotenuse?
Correct answer: D
The factor from 171717 to 858585 is 555. Multiply the original hypotenuse by that same factor.
Which pair of parameters guarantees that the generated triple is primitive?
Correct answer: B
The pair 141414 and 555 has opposite parity and no common prime factor.
The pair 141414 and 777 shares 777. The pair 151515 and 777 is odd in both entries. The pair 121212 and 888 shares 444.
Why is the output for parameter pair (13,7)(13,7)(13,7) a scaled triple?
Both parameters are odd. Their square difference and square sum are even, and the doubled product is even too.
The parameters 101010 and 444 share a factor of 222. What factor is guaranteed in all three entries because of that shared factor?
Squaring either input, or multiplying the two inputs, introduces two copies of the shared factor.
The doubled product is 808080, but the square difference is 848484, so 161616 does not divide all entries.
Which input falls outside the domain for positive Pythagorean triples because it makes the doubled product zero?
The domain requires m>n>0m>n>0m>n>0. For n=0n=0n=0 the doubled product gives a zero leg.
A primitive triple has odd leg 191919 and hypotenuse 181181181. What is the smaller parameter square?
The smaller square is the half-difference of the hypotenuse and odd leg.
A list records whether the odd or even length occurs at each side of a right triangle with integer lengths. Which list can describe a primitive triple?
In a primitive triple, there is exactly one even leg. The sum of an odd square and an even square is odd.
The hypotenuse therefore has an odd square and is itself odd.
The doubled product in Euclid's formulas is 525252 and n=2n=2n=2. What is mmm?
Put 222 in place of nnn and solve the equation.
When divided by 444, an odd square leaves remainder 111. What remainder would the sum of two odd squares leave?
Each odd square contributes one beyond a multiple of 444.
This is why two odd legs cannot give a square hypotenuse.
A triple's primitive sides are 282828, 195195195, and 197197197. The corresponding scaled triangle has longest side 1,3791{,}3791,379. What is the GCF of the scaled lengths?
Divide the two corresponding hypotenuse lengths to get the scale factor.
The original side lengths have GCF 111, so the scaled lengths have GCF 777.
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