To find the least integer whose square root is an irrational number between 5 and 7, we first need to determine the range of numbers whose square roots fall within that interval.
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The lower bound is the square of 5: \[ 5^2 = 25 \]
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The upper bound is the square of 7: \[ 7^2 = 49 \]
Hence, we need to find an integer \( n \) such that: \[ 25 < n < 49 \]
Next, we want \( n \) to be an integer that is not a perfect square (because perfect squares have rational square roots). The perfect squares between 25 and 49 are:
- 25 (which is \( 5^2 \))
- 36 (which is \( 6^2 \))
- 49 (which is \( 7^2 \))
The integers between 25 and 49, excluding 25 and 36 (which are perfect squares), are 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48.
The smallest integer in this list is 26.
Thus, the least integer whose square root is an irrational number between 5 and 7 is: \[ \boxed{26} \]