37
36
Prime number between 196 and 289 whose digits add to 14? Possibles 239, 248, 257, 266, 275, 284. 248, 266 and 284 are multiples of 2, 275 is a multiple of 5. This leaves 239 and 257. Only 239 has one digit which is the square of another. So there you have it.
Square them both, find a non-square integer between those two results, and then take the square root of that number. In other words, find a non-square integer between 25 and 49, and since there is only one square number between them, 36, that should be easy; let's pick 42, and then take the square root of it. Ta da! √42 is an irrational number between 5 and 7, its first 30 digits being 6.48074069840786023096596743608.
9
81
36
144
36=6*6=2*18=9*4.
Such a number does not exist. 3600 is 60 squared and 4900 is 70 squared There is no number between 60 and 70 that is a multiple of 15 so there can be no square between 3600 and 4900 that is a multiple of 15.
There can't be a prime number that has a square root because the square root would be a factor of the number.
289
The number nine is a composite number (3 x 3 = 9) and it is a square number (32 = 9). It is a divisor of any number that has digits having the sum of nine or a multiple of 9 (For example, the digits in 367,218 add up to 27, which is a multiple of 9, and the digits in 27 also add up to 9)
144
64
4225
4 and 16 both qualify.
Prime number between 196 and 289 whose digits add to 14? Possibles 239, 248, 257, 266, 275, 284. 248, 266 and 284 are multiples of 2, 275 is a multiple of 5. This leaves 239 and 257. Only 239 has one digit which is the square of another. So there you have it.