To generate a new Pythagorean triple by multiplying each integer in the given triple (20, 21, 29) by 5, we perform the following calculations:
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Multiply 20 by 5: \[ 20 \times 5 = 100 \]
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Multiply 21 by 5: \[ 21 \times 5 = 105 \]
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Multiply 29 by 5: \[ 29 \times 5 = 145 \]
So, the new Pythagorean triple is (100, 105, 145).
To verify that (100, 105, 145) is indeed a Pythagorean triple, we can check if \(100^2 + 105^2 = 145^2\):
Calculating \(100^2\): \[ 100^2 = 10000 \]
Calculating \(105^2\): \[ 105^2 = 11025 \]
Adding these two results: \[ 10000 + 11025 = 21025 \]
Now calculating \(145^2\): \[ 145^2 = 21025 \]
Since \(100^2 + 105^2 = 145^2\), we confirm that (100, 105, 145) is a valid Pythagorean triple.
Thus, the Pythagorean triple generated by multiplying (20, 21, 29) by 5 is: \[ \boxed{(100, 105, 145)} \]