Question 27

We will take three common primitive Pythagorean triples, generate five scaled versions for each, and then check if these new triples are primitive. A Pythagorean triple $(a, b, c)$ is a set of three positive integers that satisfy the equation $a^2 + b^2 = c^2$. A primitive Pythagorean triple has no common factors other than 1. This means their greatest common divisor (GCD) is 1. We will use the primitive triples (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Step 1 — Scaled Versions for (3, 4, 5)

The triple (3, 4, 5) is primitive because the greatest common divisor of 3, 4, and 5 is 1. Let us generate 5 scaled versions by multiplying each number by an integer $k$. We will use $k = 2, 3, 4, 5, 6$.

For $k=2$: The scaled triple is $(2 \times 3, 2 \times 4, 2 \times 5)$. $= (6, 8, 10)$ The common factors of 6, 8, and 10 are 1 and 2. The greatest common divisor (GCD) is 2. Since the GCD is 2 (not 1), this triple is not primitive.

For $k=3$: The scaled triple is $(3 \times 3, 3 \times 4, 3 \times 5)$. $= (9, 12, 15)$ The greatest common divisor (GCD) of 9, 12, and 15 is 3. Since the GCD is 3 (not 1), this triple is not primitive.

For $k=4$: The scaled triple is $(4 \times 3, 4 \times 4, 4 \times 5)$. $= (12, 16, 20)$ The greatest common divisor (GCD) of 12, 16, and 20 is 4. Since the GCD is 4 (not 1), this triple is not primitive.

For $k=5$: The scaled triple is $(5 \times 3, 5 \times 4, 5 \times 5)$. $= (15, 20, 25)$ The greatest common divisor (GCD) of 15, 20, and 25 is 5. Since the GCD is 5 (not 1), this triple is not primitive.

For $k=6$: The scaled triple is $(6 \times 3, 6 \times 4, 6 \times 5)$. $= (18, 24, 30)$ The greatest common divisor (GCD) of 18, 24, and 30 is 6. Since the GCD is 6 (not 1), this triple is not primitive.

Step 2 — Scaled Versions for (5, 12, 13)

The triple (5, 12, 13) is primitive because the greatest common divisor of 5, 12, and 13 is 1. Let us generate 5 scaled versions using $k = 2, 3, 4, 5, 6$.

For $k=2$: The scaled triple is $(2 \times 5, 2 \times 12, 2 \times 13)$. $= (10, 24, 26)$ The greatest common divisor (GCD) of 10, 24, and 26 is 2. Since the GCD is 2 (not 1), this triple is not primitive.

For $k=3$: The scaled triple is $(3 \times 5, 3 \times 12, 3 \times 13)$. $= (15, 36, 39)$ The greatest common divisor (GCD) of 15, 36, and 39 is 3. Since the GCD is 3 (not 1), this triple is not primitive.

For $k=4$: The scaled triple is $(4 \times 5, 4 \times 12, 4 \times 13)$. $= (20, 48, 52)$ The greatest common divisor (GCD) of 20, 48, and 52 is 4. Since the GCD is 4 (not 1), this triple is not primitive.

For $k=5$: The scaled triple is $(5 \times 5, 5 \times 12, 5 \times 13)$. $= (25, 60, 65)$ The greatest common divisor (GCD) of 25, 60, and 65 is 5. Since the GCD is 5 (not 1), this triple is not primitive.

For $k=6$: The scaled triple is $(6 \times 5, 6 \times 12, 6 \times 13)$. $= (30, 72, 78)$ The greatest common divisor (GCD) of 30, 72, and 78 is 6. Since the GCD is 6 (not 1), this triple is not primitive.

Step 3 — Scaled Versions for (8, 15, 17)

The triple (8, 15, 17) is primitive because the greatest common divisor of 8, 15, and 17 is 1. Let us generate 5 scaled versions using $k = 2, 3, 4, 5, 6$.

For $k=2$: The scaled triple is $(2 \times 8, 2 \times 15, 2 \times 17)$. $= (16, 30, 34)$ The greatest common divisor (GCD) of 16, 30, and 34 is 2. Since the GCD is 2 (not 1), this triple is not primitive.

For $k=3$: The scaled triple is $(3 \times 8, 3 \times 15, 3 \times 17)$. $= (24, 45, 51)$ The greatest common divisor (GCD) of 24, 45, and 51 is 3. Since the GCD is 3 (not 1), this triple is not primitive.

For $k=4$: The scaled triple is $(4 \times 8, 4 \times 15, 4 \times 17)$. $= (32, 60, 68)$ The greatest common divisor (GCD) of 32, 60, and 68 is 4. Since the GCD is 4 (not 1), this triple is not primitive.

For $k=5$: The scaled triple is $(5 \times 8, 5 \times 15, 5 \times 17)$. $= (40, 75, 85)$ The greatest common divisor (GCD) of 40, 75, and 85 is 5. Since the GCD is 5 (not 1), this triple is not primitive.

For $k=6$: The scaled triple is $(6 \times 8, 6 \times 15, 6 \times 17)$. $= (48, 90, 102)$ The greatest common divisor (GCD) of 48, 90, and 102 is 6. Since the GCD is 6 (not 1), this triple is not primitive.

Answer

(i) For the primitive triple (3, 4, 5): The 5 scaled versions are (6, 8, 10), (9, 12, 15), (12, 16, 20), (15, 20, 25), and (18, 24, 30). None of these scaled versions are primitive. (ii) For the primitive triple (5, 12, 13): The 5 scaled versions are (10, 24, 26), (15, 36, 39), (20, 48, 52), (25, 60, 65), and (30, 72, 78). None of these scaled versions are primitive. (iii) For the primitive triple (8, 15, 17): The 5 scaled versions are (16, 30, 34), (24, 45, 51), (32, 60, 68), (40, 75, 85), and (48, 90, 102). None of these scaled versions are primitive.