It is possible to find out using mathematical methods whether a given integer is a prime number or not.
Yes, 78 659 177 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 78 659 177, the only two divisors are 1 and 78 659 177. Therefore 78 659 177 is a prime number.
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Since 78659177 is a prime number, 78659177 is also a deficient number, that is to say 78659177 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 78659177 without 78659177 itself (that is 1, by definition!).