12 multiple-choice questions, progressively harder.
The parameters 333333 and 212121 share GCF 333. After removing that factor from both, the new parameters are both odd. What is the GCF of the original output?
Solution
Correct answer: D
Removing 333 from the inputs removes a factor 999 from the outputs. The reduced pair (11,7)(11,7)(11,7) is coprime and odd in both entries, so its output still has a factor 222.
The hypotenuse 170170170 has exactly one factor of 222. No odd prime divides all the reduced outputs, since it would divide both reduced parameters.
The diagram shows the sides of a scaled triple. Which pair generates its primitive triple?
Correct answer: A
First divide all lengths by their GCF of 555. The primitive triple has odd leg 656565 and hypotenuse 979797.
Take the positive roots.
A primitive triple is generated with n=3n=3n=3. Which of these values for mmm is possible?
Correct answer: C
Opposite parity requires an even value of mmm since nnn is odd. Coprimality rules out multiples of 333.
Of the four values, only 202020 meets both tests.
The odd leg of a primitive triple is a prime. Which relation between the parameters follows?
Factor the formula for the odd leg.
Both factors are positive integers, and the second exceeds 111. Since their product is prime, the smaller factor is 111.
A primitive triple has even leg bbb. Which of these conclusions is forced by Euclid's formulas?
Correct answer: B
Exactly one parameter is even, so the product mnmnmn is even. The formula contributes one more factor of 222.
For example, the parameter pair (22,1)(22,1)(22,1) gives b=44b=44b=44, which is divisible by 444 and has no factor of 888.
A primitive triple has an even leg of 686868. Which possibility has hypotenuse below 500500500?
The parameter product must be 343434. Its pairs with m>nm>nm>n are (34,1)(34,1)(34,1) and (17,2)(17,2)(17,2). Both meet the primitivity tests.
Only the second hypotenuse is below 500500500. The square difference for (17,2)(17,2)(17,2) gives odd leg 285285285.
In a primitive triple, the half-sum of the hypotenuse and odd leg is 256256256, and the half-difference is 494949. What is the even leg?
The positive roots of the two squares are 161616 and 777. The even leg is twice their product.
What property makes the recovered parameters unique when the triple is primitive and m>n>0m>n>0m>n>0?
The odd leg and hypotenuse fix the two values.
Each square has a unique positive root. The larger of the two roots must be mmm.
The product of two positive coprime integers has prime factorization 26⋅54⋅722^6\cdot5^4\cdot7^226⋅54⋅72. One factor contains 222 and 777 and the other contains 555. What is the smaller factor?
Coprimality means every copy of a given prime goes into just one factor.
The factor containing 555 is the smaller one.
One right triangle comes from (17,3)(17,3)(17,3) and another from (10,7)(10,7)(10,7). Which statement describes their relationship?
The first pair generates (280,102,298)(280,102,298)(280,102,298). The second generates (51,140,149)(51,140,149)(51,140,149).
Reorder the second triangle's legs to compare corresponding lengths.
For a right triangle with integer sides, a prime divides one leg and the hypotenuse. Why must it divide the other leg?
Call the given leg aaa and the other leg bbb.
The prime divides both terms on the right, so it divides their difference. A prime dividing b2b^2b2 must divide bbb.
A triple comes from coprime parameters that are both odd. Which statement gives the exact GCF of the outputs?
Both square sum and square difference are even, and the doubled product is even. The square sum has remainder 222 on division by 444.
An odd prime dividing all three would divide the square sum and difference, hence both parameter squares. Coprimality rules that out.
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