It is possible to find out using mathematical methods whether a given integer is a prime number or not.
Yes, 10 289 is a prime number.
Indeed, the definition of a prime numbers is to have exactly two distinct positive divisors, 1 and itself. A number is a divisor of another number when the remainder of Euclid’s division of the second one by the first one is zero. Concerning the number 10 289, the only two divisors are 1 and 10 289. Therefore 10 289 is a prime number.
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Since 10289 is a prime number, 10289 is also a deficient number, that is to say 10289 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 10289 without 10289 itself (that is 1, by definition!).