The smallest digit of the Pythagorean triplet is 3
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Euclid proved that if a and b are numbers with a > b, then a^2 - b^2 and 2ab and a^2 + b^2 are a Pythagorian triple.
Either a^2 - b^2 or 2ab has to be the smallest digit given. If the smallest digit is odd, it cannot be 2ab.
Try it. Say the smallest digit is 3. The only perfect squares that differ by 3 are 4 and 1. So a = 2 and b= 1. The Pythagorian triplet is 3, 4, 5.
Say the smallest digit is 5. The only perfect squares that differ by 5 are 9 and 4. So a = 3 and b = 2. The Pythagorian triple is 5, 12, 13.
Say the smallest digit is 7. The only perfect squares that differ by 7 are 16 and 9. So a = 4 and b = 3. The Pythagorian triple is 7, 24, 25.
How about at even smallest digit. Say the smallest digit is 8. The only perfect squares that differ by 8 are 9 and 1. So a = 3 and b= 1. The Pythagorian triple is 8, 6, 10. But then 8 is not smallest. So 2ab = 8, so a = 4 and b = 1. The triple is 15, 8, 17.
One more. Say the smallest digit is 10. There are no perfect squares that differ by 10. So it must be that 2ab = 10. So a = 5 and b = 1. The triple is 24, 10, 26, which is a multiple of 5, 12, 13.
Step-by-step explanation: