Pythagorean Triplet Finding
A Pythagorean triplet includes three positive integers $a, b, c$ such that $a^2 + b^2 = c^2$. We need to identify a triplet where 8 is the smallest integer.
Evaluate Triplets
We test each option against the Pythagorean theorem ($a^2 + b^2 = c^2$) and the smallest member condition.
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Option 1: 8, 15, 19
Test: $8^2 + 15^2 = 64 + 225 = 289$. $19^2 = 361$. Since $289 \neq 361$, it's not a Pythagorean triplet.
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Option 2: 8, 10, 12
Test: $8^2 + 10^2 = 64 + 100 = 164$. $12^2 = 144$. Since $164 \neq 144$, it's not a Pythagorean triplet.
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Option 3: 8, 15, 17
Test: $8^2 + 15^2 = 64 + 225 = 289$. $17^2 = 289$. Since $289 = 289$, this is a Pythagorean triplet. The smallest member is indeed 8.
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Option 4: 8, 18, 28
Test: $8^2 + 18^2 = 64 + 324 = 388$. $28^2 = 784$. Since $388 \neq 784$, it's not a Pythagorean triplet.
Result
The triplet (8, 15, 17) correctly satisfies the conditions of being a Pythagorean triplet with 8 as its smallest member.
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