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.
There can't be a prime number that has a square root because the square root would be a factor of the number.
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.
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)
64
144
4225
The number of digits in the square root of a number depends on the number. If it is a square number, the square root will have a finite number of digits. If the number is not a square number then the square root will be an irrational number with an infinite, non-repeating decimal representation. In both cases, the number of digits before the decimal point, in the square root of x will be the rounded value of 1+0.5*log10(x)
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.